Name ______________________________________________ Math 1321.02 – Quiz – Jan. 23, 2004 1. State the quadratic formula. x = 2. Factor x3 - 8 = ____________________ 3. Solve.

a. 2

3−x

= 1

4+x

→_____________ b. x2 = 4x → ____________

4. Find the x-intercept of y = x2 – 4 → ____________ y = 2x → ________________ 5. Find the y-intercept of y = x2 + 3x - 4 → _____________ y = cos x → _____________ 6. Complete each of the following triangles x = _______ a = ________ t = _________

Name ______________________________________ Math 1321 – Short Quiz – Jan. 26, 2004 1. If P ( x + 2, 3 ) and Q( 2x, y ) are ordered pairs of points with P = Q, then find the values of x and y. x = _________ y = _________ 2. Let A = { 1, 2, 3 } and B = { a, b }. Find A x B = _______________________________ 3. Draw a rectangular coordinate syste. a) label the axis b) label the quadrants 4. Plot the points A ( 2, - 4 ), B ( 0, 5 ), and C(- 4, 0 ). 5. Find each of the following absolute values. Exact values. a) | 5 - 13 | = ____________ b) | 4 - π | = _____________ 6. Use the distance formulas )12()12( yyxx −+−=d to find the distance between A(2, -3) and B(4, 1) Exact answer is required.

Name __________________________________________ Math 1321- Short Quiz #3 – Jan. 28, 2004 1. Write the distance formula to find the distance between the points A(a, b ) and B ( c, d ). Use the given letters in your formula. d = 2. Use the distance formula a) to find the value of x if P( x, 2 ) is a point that is equidistant from A( 4, 1) and B( 1, - 3 ) . → x = _________ b) to show that A(2, -1), B( 3, 0 ), and C ( 5, 2) are collinear. c) to show that the points A( 3, - 2 ), B( 3, 0 ) , and C( -2, 0 ) form a right triangle.

3. Find the determinant of matrix A if A = . → det A = _____________

−5234

Name ___________________________________________ Math 1321- Quiz #4 Feb. 2, 2004 1. Write the formula for finding the division of a line segment if A(x1, y1) and B(x2, y2) are the endpoints of the segment with P(x,y) being the point that divides the segment and r is defined as the directed distance r = AP/AB. x = _________________________ y = _____________________ 2. What is the midpoint of the segment A(2, -3) and B(4, -1 ) ? ___________________________ 3. Find the value of r as defined in #1 if P is inside the segment and P a) is three times as far from A as P is from B. _______________ b) if P is three times as far from A as P is from B and P is outside the segment AB extended through B. ____________ 4. A man is three-fifths of the way from the tip of his shadow to a light post (along the ground). If his shadow is 9 feet long and he is 6 feet tall, then how high is the light pole (light)? 5. How would the coordinates below have to be related so that the parallelogram represents a rhombus? Use the coordinates and algebra to answer this – make sure your answer is not based solely on words.

Name _______________________________________________ Math 1321 – Quiz #5 – Feb. 6, 2004 1. A function f(x) is said to be ________________________ if f( - x ) = f(x). 2. True or False. _______________ a) The sum of two odd functions is odd. _______________ b) The product of two odd functions is even. _______________ c) the diagonals of a rectangle are congruent (equal in length) _______________ d) all rectangles are parallelograms _______________ e) triangles that are congruent to each other are also similar to each other

3. Let 42

3)(+

=x

xxf . Find

a) f( 0 ) . _________________ b) f( h ) = _____________ c) The domain consists of all real numbers except what value of x ? x = ____________ 4. P(x,y) is twice as far from A(2, -1) as it is from B ( 0, 5 ). Write an equation that expresses all points that satisfy that condition. 5. Show whether the following functions are even, odd, or neither

a) f(x) = 3 b) f(x) = 2 2x

Name ______________________________________________ Math 1321- Quiz # 6, Feb. 11, 2003 1. Complete the following right triangles C B A C B A side a = 24, side c = 24 2 side a = 12, angle B = 600 find angle B→ ________ and side b _______ find side b = ________ and side c. ________ 2. True or False a)The following represents the graph of y = sin x . → __________ b) the following represents the graph of y = 2x + 4 → ____________ 3. x2 - 4 = 0 can only be zero when x = 2. True or False. → _________________ 4. To find the length of a diagonal of a given rectangle, you use ? _______________________ To compare that two diagonals are of equal length you would ? 5. If you are trying to prove that two segments are perpendicular, then you can use the fact that the product of their slopes = -1. True or False. → ________________ 6. The easiest and quickest way to determine if three points are collinear is to → 7. If P is a point that is 12 units away from point A and P happens to 4 units away from point B with P on the segment AB extended through B, then what is the value of r ( in the formula for the division of a line segment). | P x = x1 + r(x2-x1) r = ________ | B A |

Name _________________________________________ Math 1321 – Quiz # 7, Feb. 13, 2004 1. The domain of the function y = Arcsin x is all x/ such that -1 ≤ x ≤ 1 while the range is -π/2 ≤ y ≤ π/2 What are the domain _________________ and the range _________________ of the function y = Arccos x. 2. If f(x) = 2x, and g(x) = ( x – 2 ) 2, then find f og (1) = f(g(1)). ____________ 3. If f(x) = 2x – 4 and g(x) = (x+4)/2, find g o f( 13 ). ____________ 4. Find the slope of each of the following lines a) a line that is perpendicular to the line passing through the segment AB, A(3, -2) and B(-3, 4) . → __________ b) a line that is parallel to the line x = 3 . → _______________ c) x/2 - y/3 = 1 → _______________ 5. What is the equation of the line that a) has slope 2 and crosses the y-axis at the point ( 0, - 4 ) → _________________ b) has no slope ( undefined slope) and passes through the point ( 2, 5) → ____________ c) is parallel to the line x + 2y = 3 and passes through the origin. → _____________ d) is the perpendicular bisector of the segment (2, -1) and (4, 1) → ____________

Name _________________________________________ Math 1321 – Quiz #8 , Feb. 16, 2004 1. Find the solution of the following system of equations by using the elimination method. ax + y = c x – by = d Solve for x only. x = ___________ 2. Use the substitution method to solve for y ( solve only for x) x2 - y2 = 4 ( hyperbola ) x – y = 3 ( line ) 3. Find the directed distance from the line 4x – 3y = 6 to the point ( -1, 0 ) 4. What is the diameter of a circle with center at the point ( -1, 0) and tangent to the line 4x – 3y = 6 5. Two pipelines are parallel to each other. If they are 4 meters apart and the first pipeline can be represented by the equation 12x + 5y = 12 Find the equation of the pipeline if the pipeline you are looking for is above the one represented by 12x + 5y = 12 (HINT: the line you are looking for is of the form 12x + 5y + K = 0 and is 4 units away from the line 12x + 5y = 12 ) there are two possible answers – depending on whether you are trying to find the line above or below

Name ________________________________________________ Math 1321 – Quiz #9 – Feb. 18, 2004 1. Describe each of the following curves by giving a complete description. Include things such as slope, center, radius, or the exact description(1d). a) y = - 2 → _____________________________________________________________________ b) y = x/3 – 6 → ________________________________________________________________ c) (x – 2 ) 2 + ( y – 4 )2 = 9 → ________________________________________________________ d) x2 + y2 = 0 → _______________________________________________ y-axis 2. Graph the following curve. (x+2)2 + y2 = 9 x-axis 3. What is the equation of the circle with center at (2. -5 ) and radius = 7. __________________________________ 4. Why is ( x + 2)2 + ( y -1)2 = - 4 not a circle ? __________________________________________________ 5. What is the equation of the circle with center at (2, 3) and passing through the point (-1, - 1 ) ? ____________________

Name __________________________________________________ Math 1321 – Quiz #10 – Feb. 20, 2004 1. Write the following in the general form of a circle. (x +2)2 + (y-2)2 = 4 2. Write in the standard form of a circle. x2 + 4x + y2 – 2y – 4 = 0 3. What is the equation of the circle with a) center at ( 3, -2 ) and tangent to the line 3x – 4y = 2. b) If the circle has its center on the x-axis and also on the line 4x – y = 8 and has a diameter of length 12, then find the equation.

Name _________________________________________ Math 1321- Qz #11 – Feb. 24, 2004 1. If f , f(x) , represents a constant function, then f(x) must be nondecreasing and nonincreasing. Pf. Begin with function f being a constant function, f(x) = c. Prove that it is nondecreasing and nonincreasing. Let x1 and x2 be given with x1 < x2. a) If f is increasing, then f(x1) < f(x2), but f(x1) = ____ and f(x2) = ____. So, f(x1) _______ f(x2) which means f can not be increasing → f is nonincreasing. Also, b) If f is decreasing, then f(x1) > f(x2), but we know that f(x1) = _____ and f(x2) = ______ f decreasing means that f(x1) > f(x2), but in this case we have ________________ We conclude that f is 2. Suppose that a pipe leans against a wall and a cross section has the appearance below. Find the equation of the circle represented by the cross section. The circle is tangent to both sets of axes and is of radius 4. The line L has slope crosses the y-axis at y=12 and the x axis at x = 16. (Similar but easier than #35 on page 75) C: __________________________ 3. Find an angle created by the intersection by the intersection of the lines φ = ______________ (Short and Sweet) Should not need more space than what is provided 4x – 3y = 4 3x + 4y = 1 4. Find the acute angle bisector created by the intersection of the lines equation1: 3x – 4y = 3 equation2: 4x – 3y = 4 2 1

Name _________________________________________ Math 1321- Qz #12 – Feb. 25, 2004 1. If f , f(x) , represents a constant function, then f(x) must be nondecreasing and nonincreasing. Pf. Begin with function f being a constant function, f(x) = c. Prove that it is nondecreasing and nonincreasing. Let x1 and x2 be given with x1 < x2. a) If f is increasing, then f(x1) < f(x2), but f(x1) = ____ and f(x2) = ____. So, f(x1) _______ f(x2)

which means f can not be increasing → f is nonincreasing. Also, b) If f is decreasing, then f(x1) > f(x2), but we know that f(x1) = _____ and f(x2) = ______ f decreasing means that f(x1) > f(x2), but in this case we have

________________ We conclude that f is 2. Suppose that a pipe leans against a wall and a cross section has the appearance below. Find the equation of the circle represented by the cross section. The circle is tangent to both sets of axes and is of radius 4. The line L has slope crosses the y-axis at y=12 and the x axis at x = 16. C: __________________________ 3. Find an angle created by the intersection by the intersection of the lines φ = ______________ 4x – 3y = 4 3x + 4y = 1 4. Find the acute angle bisector created by the intersection of the lines

equation1: 3x – 4y = 3 d1 = distance from line L1 to P(x,y) = 22 )4(3

3)(4)(3

−+−

−− yx

equation2: 4x – 3y = 4 2 1

d2 = distance from lineL2 to P(x,y) = 22 )3(4

4)(3)(4

−+−

−− yx

Name ___________________________________________ Math 1321 – Quiz #13, Feb. 27, 2004 1. Write down the translation formulas when the origin is translated to the point O/ (2, -1 )

( there should be no h or k in your formla) 2. Find the new coordinates of P(2, -4) when the origin is moved to O/ ( -2, 3) . __________________ What will the equation x + 2y = 3 look like in terms of x/ and y/ after the translation ? __________________________________ 3. Find the point to which the origin must be translated to so that the new equation ( in terms of x/ and y/) a) (x -2)2 + ( y+5)2 = 9 will be free of all first degree terms, what does the equation look like ? point: ______________ equation: _____________________ b) Same as above with the hyperbola (x -2)2 - ( y+ 3)2 = 4 point: _______________ equation: _____________________ 4. Find the point to which the origin must be translated to so that the new equation will be free of one first degree term

(just one) and the constant term. x2 - 4x = 2y - 6 point: _________________ equation: ____________

Name ______________________________________________ Math 1321 – Quiz #14, March 1, 1. Identify each of the following by using the term that best describes the graph of the equation point (what point), vertical line, horizontal line, slant line, circle, point, no graph,

a parabola that opens to the right-left-up-down a) (x+2)2 + (y-3) = - 1 → __________________________________ b) y = - 2 → _____________________ c) y2 = ( x + 2 ) → ________________ d) x2 + y2 + 2x + 4y - 400 = 0 → ____________________ e) x2 – 2x + y - 1 = 0 → _______________________ f) ( x+2 )2 + ( y – 1 )2 = 1 → ___________________________ 2. Find the domain of a) ( x +1)2 + y2 = 1 → _________________________________________ b) y2 = x + 1 → ___________________________________________ c) x = 3 → _________________________________ 3. Find the range of a) y2 = x+ 1 → _____________________________________ b) x2 + (y-1)2 = 1 → ________________________________ 4. Find the equation of the parabola with a) vertex at the origin and focus at F(-2, 0 ). → _________________________________ b) directrix y = -3 and Focus at F(0, 3) → _________________________________

Name ______________________________________ Math 1321 – Quiz #15, March 3, 2004 1. Write down the general form of a parabola that opens to the sides ( left, right) 2. Write down the standard form of a parabola that opens up or down. 3. What is the equation of the parabola with a) focus at ( 2, -3) and vertex at V( ( 0, - 3 ) b) l.r. of length 12, opens down with vertex at ( 2, - 5) c) passes through the point ( 2, 0 ) and axis of symmetry parallel to the x-axis with vertex V(1, 4) 4. Symmetry: a) a curve is said to be symmetric about the x-axis if (x,y) is a point on the curve, then so is the point

____________ b) If for every point P(x,y) on a given curve the point Q( -x, - y) is also on the curve, then what type of symmetry does this curve have ?

Name ______________________________________ Math 1321- Short Quiz -#16 – March 8, 2004 1. Write the standard form of an ellipse with center at (h,k) and minor axis parallel to y-axis. 2. Write the standard form of a hyperbola with transverse axis (major) parallel to y-axis and center at (h,k) 3. Find the equation of an ellipse whose sum of the distances from two fixed points is 60 and endpoints of minor axis are at a) B(3, -2 ) and B/ ( - 7, - 2) b) we defined b2 = ___________________ ( for an ellipse ) c) what is the eccentricity of this ellipse ? e = c/a = _______________

4. The equation of a hyperbola is given by 114

)2( 22

=−+ yx

. Find

a) length of the conjugate (minor) axis . _____________ b) length of the l.r. ( 2b2/a ) . _______________ c) we defined b2 = _______________ e) asymptotes: in the form ax + by = c 5. A satellite travels in an elliptical path around a planet. If the farthest the satellite gets from the planet is 180 units and the closest it gets is 20 units,units then find an equation that represents the orbit of the satellite. Assume that the planet is a f fixed point.

Name ________________________________________________ Math 1321 – Quiz #17, March 24, 2004 1. We finished studying conics. List the three conics we studied ( circles could be included – but the other three would be ?) 2. Find the x-intercepts of a) f(x) = x2 – 2x → _________________

b) y = 22

−+

xx

→ ____________________

3. What are the y-intercepts of a) f(x) = (x+2) (x – 3 ) → _________________ b) y = - 2 → _______________

c) y = 22

−+

xx

→ ____________________

4. What are the vertical asymptotes of

a) y = 22

−+

xx

→ ____________________

b) y = 92 −x

x → ________________

5. What are the horizontal asymptotes of

a) y = 222−+

xx

→ ____________________

b) y = 92 −x

x → ____________________

Name ______________________________________________________ Math 1321 – quiz #18, March 26, 2004 1. Sketch the graph of y = x – 4

2. Sketch the graph of f(x) = 4162

+−

xx

x-intercept: _________ y-intercept: ____________ V.A. ___________ H.A. _________

3. Sketch the graph of y = xx

xx2

322

2

+−−

x-intercept: _________ y-intercept: ____________ V.A. ___________ H.A. _________

Name _____________________________________ Math 1321 – Quiz #19, 2004 1. Find the vertical, horizontal, and vertical asymptote(s) of

b) y = 4

342

−+−

xxx

V. A. : _________________ H.A. : ___________________ S.A.: ______________ b) xy + y – x2 + 1 = 0 → y = V. A. : _________________ H.A. : ___________________ S.A.: ______________

2. Given y = 1

432

2

+−−

xxx

, we find that y = 1 is a horizontal asymptote.

Does the the curve cross it- if so, at what point P(x,y). Show algebraically-do not need a graph.

3. Given y = 2

322

−−+

xxx

we get y = 2

322

−−+

xxx

= x + 4 - 2

11−x

What is the slant asymptote ? _____________

Does the curve cross it ? replace y with y = x +4 and solve for x. →y = 2

322

−−+

xxx

Name _______________________________________________ Math 1321 – Quiz #20, April 2, 2004 1. Define y = cosh x ( not in words but in terms of its mathematical definition – equation ) cosh x = 2. Sketch the graph of y = sinh x. You need to draw the asymptotes to this function and label them. 3. Find 2sinhx • cosh x by using their definitions and simplifying ( with trig functions- 2sinx cos x = sin 2x. 4. Find x a) x = - 24 → x = ____________ b) log ½ 8 = x → x = ____________ c) If log

3 y = 0.24, then what does log 3 y5 = x ? x = ____________ 5. What is the domain of y = log 4 x ? ________________________________________ 6. What is the range of y = - 3 x

→ __________________________

7. If f(x) = 2x – 6 and g(x) = 2

6+x and you know that f and g are inverses of each other, then what does

f o g (r) = ? _________

Name _______________________________________ Math 1321 – Short Quiz # 21 (?) - April 5, 2004 1. Graph the points ( polar coordinates ) a) P ( 3, - 200o ) θ=90o b) Q( -2, - π/4 ) θ=180o polar axis θ = 270o 2. Write down the formulas to change from rectangular to polar coordinates tan θ = ____________ r2 = _______________ 3. Write down the formulas to change from polar to rectangular x = ____________ y = __________ 4. Change to polar coordinates a) ( 3, - 3 ) → ______________ with | θ | < 360o b) ( )34,4 → _____________ with | θ | < 360o c) ( 0, - 2 ) → ______________ with | θ | < 2π 5. Change to rectangular coordinates ( exact values ) a) ( 4, - 60o ) → x = _______ y = ________ b) ( -4, - 3π/2) → x = _______ y = ________

Name ______________________________________ Math 1321 – Quiz 22– April 14, 2004 1. In the equation y = mx + 2, how many different complete curves ( more than just a point ) can be generated by letting m=2 In the equation x = 2t2 , y = t, how many different curves(more than just a point) can be generated by letting t = 1. 2. What kind of curve does each of the following polar equations represent a) r = 4 sin 4 θ → ____________________ b) r2 = 9 cos 2 θ → ______________________ c) r = 5 - 3 sin θ → ______________________ 3. There are three types of limacons → curve around the origin, curve with an inner loop, what the other one ? 4. How many petals does the rose curve r = 4 cos 4θ 5. Give me quick graph of r2 = 4 sin 2θ

Name _____________________________________ Math 1321 – Quiz # 23, April 16, 2004 1. Give me a parametric equation of each of these two a) 2x – 3y = 6 → b) y = x2 + 2x – 4 → ______________________________ ___________________________ 2. Eliminate the parameter in the equation(s) and write the equation in terms of x and y a) x = 2t - 2 → _________________________ y = t + 1

b) x = 2+t

t, y =

23+tt

→ ____________________

c) x = 2t, y = 4t → _______________________ d) x = sinh t, y = cosh t ( HINT: cosh2 x - sinh 2 x = ? ) → _____________________ 3. Sketch the actual graph of the given parametric equations. a) x = sin2 t, y = cos 2 t b) x = sin2 t , y = cos t

Name __________________________________ Math 1321 – Quiz #24 --- April 21, 2004 1. Plot the points A(3, 4, 0 ) and B( 4, 7, 9 ) 2. Find the midpoint of A and B above. 3. Find the distance between A and B above. 4. Find a point P that is twice as far from A as from B and is between A and B ( along the segment AB). 5. Sketch the graph of a) x + 2y + 3z = 6 b) x + 2z = 4

Name __________________________________________ Math 1321 – Quiz #26 -- April 26, 2004 1. Draw and label a vector --- tail, head 2. Use the two given vectors to find the resultant vector ( A + B ) . 3. Construct a parallelogram by using the two given vectors in their current position to describe (draw) the difference of A and B ( A – B). 4. Find a vector of length 1 having opposite direction of the positive x-axis. ________________ 5. Find any vector that has length 8. ___________________ 6. What do we call a vector of length 1 ? _____________________ 7. What is the length of the vector -3i - 4j ? | -3i - 4j | = ______________

Name _______________________________ Math 1321 – Quiz # 27 – April 28, 2004 1. Let vectors A and B be points on the plane with A = ( 2, 4 ) and B = (- 3, 4 ) ( write your answers in the form ai + bj ) - Find a vector

from the origin to point A from the origin to point B → __________ → ___________ A and B below represent the vectors from the origin to the points A and B, respectively a) Find A + B = ______________ b) Find A – B = ______________ c) Find 3A = _________________ d) Find a vector of length 1 having the same direction as vector A e) Find a vector from A to B. 2. Given that vectors A and B are parallel but not equal.

Relate A and B in terms of an equation. →

3. If P is the midpoint of the segment AB, and OA and OB represent vectors from the origin to A and B respectively, then Write an equation for OP in terms of the other vectors. OP = ____________________ 4. In class we defined the dot product (scalar product ) by A • B = | A | | B | cos θ, where 0 ≤ θ ≤ 180o If A • B = 0 and A and B have nonzero lengths, then what would θ equal ? ____________ What does that say about the relationship between A and B ?

Name _______________________________ Math 1321 – Quiz # _____________ -- May 5, 2004 1. Find the dot product of

A= 3i – 2j + k and B = i – 4j + 2k . → __________ Is A • B = B • A ( commutative )? __________ 2. If A = < 2, 1, 1 > and B = < 1, 2, c > what does c have to equal so that A and B represent perpendicular vectors. c = _________ 3. What is the angle θ between vectors A and B ( from # 1 ) ? ______________________ 4. If A = 2i – 3j + k, find a vector B that twice as large as A and has the same direction. 5. Find the equation of the plane that is perpendicular to the vector N = 2i – 3j + 5k and passes through the point (1, 1, 0 ) . → ___________________________ 6. Find one vector that is perpendicular to the plane 2x – y + 5k = 1 . → _________________________ 7. What is the equation of the plane that is parallel to the plane 2x + 3y + 3z = 2 and passes through the origin ? → ________________ 8. What is the determinant of

−−−

−

440332

211

Name __________________________________ Math 1321 – Quiz --- May 7, 2004 1) If C = A • B, what can you say about C ? C is what type of quantity ? _____________ 2) mA ? What type of quantity ? ____________________________ 3) A x B = B x A ? __________________ 4 ) If C = A x B, then what can you say about C ? a) C is what kind of quantity ? _________________________________ b) magnitude: __________________ c) direction: ________________________ 5) What is the equation of a line that a) is parallel to 3i + 4j – k and passes through the point ( 3, 1, 1 ) ? b) passes through the points A( 4, 0, 2) and B( -1, 2, 1 ) ? 6. Find A x B if A = 2i – j – k and B = 4i + 2j + k .

Answers: Quiz #1:

1. State the quadratic formula. x = acbb 42 −±− ----------------------- 2a 2. Factor x3 - 8 = ____________________ → ( x )3 - (2)3 = ( x – 2 ) ( x2 + 2x + 4 ) 3. Solve.

a. 2

3−x

= 1

4+x

→_____________ b. x2 = 4x → ____________

ans. 3(x + 1) = 4(x – 2 ) ans. x2 - 4x = 0 3x + 3 = 4x – 8 x( x – 4) = 0 - x = - 11 → x = 11 x = 0 or x = 4 4. Find the x-intercept of y = x2 – 4 → ____________ y = 2x → ________________ ans. Let y = 0 → 0 = x2 – 4 Ans. None – it never crosses the x-axis → x2 = 4 → x = 2 ± 5. Find the y-intercept of y = x2 + 3x - 4 → _____________ y = cos x → _____________ ans. Let x = 0 ans. From the graph of y = cos x → y = - 4 y = 1. 6. Complete each of the following triangles x = _______ a = ________ t = _________ 72 = 52 + x2 → x = 24 = 2 6 Since legs are of equal length (6) Since one leg (8) is half → you have a 45o, 45o, 90o triangle the length of the hypotenuse a = 45o (16) → t = 30o Quiz #2

1. If P ( x + 2, 3 ) and Q( 2x, y ) are ordered pairs of points with P = Q, then find the values of x and y. x = _________ y = _________ Ans. x + 2 = 2x → x = 2 and y = 3 2. Let A = { 1, 2, 3 } and B = { a, b }. Find A x B = _______________________________ Ans. { ( 1, a), (1, b), (2,a),(2,b),(3,a),(3,c) } 3. Draw a rectangular coordinate system. a) label the axis b) label the quadrants y-axis Ans. QII QI x-axis QIII QIV 4. Plot the points A ( 2, - 4 ), B ( 0, 5 ), and C(- 4, 0 ). Ans. A is in Quadrant IV, B is on the positive x-axis, C is on the negative x-axis 5. Find each of the following absolute values. Exact values. a) | 5 - 13 | = ____________ b) | 4 - π | = _____________ ans. ) | 5 - 13 | = 5 - 13 ans. | 4 – π | = 4 – π 6. Use the distance formulas )12()12( yyxx −+−=d to find the distance between A(2, -3) and B(4, 1) Exact answer is required. ___________________ ______ __ __ Ans. dAB = \/ ( 2-4)2 + ( - 3 – 1)2 = \/ 4 + 16 = \/ 20 = 2 \/ 5 Name __________________________________________ Math 1321- Short Quiz #3 – Jan. 28, 2004

1. Write the distance formula to find the distance between the points A(a, b ) and B ( c, d ). Use the given letters in your formula. d = look it up in your notes 2. Use the distance formula a) to find the value of x if P( x, 2 ) is a point that is equidistant from A( 4, 1) and B( 1, - 3 ) . → x = _________

dAP = dBP → 22 )1()4( −+− yx = 22 )3()1( ++− yx

→ ( x-4)2 + ( y -1)2 = (x – 1)2 + ( y-3)2 remember y = 2 so → x2 – 4x + 4 + 1 = x2 – 2x + 1 + 1 → - 2x = 3 → x = - 3/2

b) to show that A(2, -1), B( 3, 0 ), and C ( 5, 2) are collinear.

dAB = 22 )01()32( −−+− dAC = 22 )21()52( −−+− dBC = 22 )20()53( −+− = 2 = 18 = 3 2 = 32 → ???? c) to show that the points A( 3, - 2 ), B( 3, 0 ) , and C( -2, 0 ) form a right triangle. do as above, then use Pythagorean theorem to show that a2 + b2 = c2.

3. Find the determinant of matrix A if A = . → det A = _____________ ans. -20 – 6 = - 26

−5234

Quiz #4 – Solutions

1. Write the formula for finding the division of a line segment if A(x1, y1) and B(x2, y2) are the endpoints of the segment with P(x,y) being the point that divides the segment and r is defined as the directed distance r = AP/AB. x = _________________________ y = _____________________ ans. LOOK in your notes 2. What is the midpoint of the segment A(2, -3) and B(4, -1 ) ? ___________________________ ( 2 + 4)/2 = x y = ( - 3 + (-1) ) /2 → P(x,y) = (3, -2) 3. Find the value of r as defined in #1 if P is inside the segment and P a) is three times as far from A as P is from B. _______________ ans. r = ¾ b) if P is three times as far from A as P is from B and P is outside the segment AB extended through B. ____ ans. r = 3/2 4. A man is three-fifths of the way from the tip of his shadow to a light post (along the ground). If his shadow is 9 feet long and he is 6 feet tall, then how high is the light pole (light)? 9/x = 6/h = 3/5 → h = 10 feet 5. How would the coordinates below have to be related so that the parallelogram represents a rhombus? Use the coordinates and algebra to answer this – make sure your answer is not based solely on words. A(0, 0 ), B(a, 0), C(a+b, c), D(b,c) My mistake on this problem: I labeled B incorrectly – B should have been B( a, o) instead of (a, b) Although the mistake is obvious from my part , it’s still possible you misunderstood. Make sure to correct and notice the answer that I wanted. ans. all sides must be the same length , say side = a → AD is of length a

→ a2 = b2 + c2 ( need a picture of a right triangle on the parallelogram to indicate this. Quiz #5

Solutions 1. A function f(x) is said to be ________________________ if f( - x ) = f(x). ans. even 2. True or False. _______________ a) The sum of two odd functions is odd. ans. true _______________ b) The product of two odd functions is even. ans. true _______________ c) the diagonals of a rectangle are congruent (equal in length) ans. true _______________ d) all rectangles are parallelograms ans. true _______________ e) triangles that are congruent to each other are also similar to each other ans. true

3. Let 42

3)(+

=x

xxf . Find

a) f( 0 ) . _________________ b) f( h ) = _____________

ans. 0 ans. 3h/ (2h+4) c) The domain consists of all real numbers except what value of x ? x = ____________ ans. x = -2 4. P(x,y) is twice as far from A(2, -1) as it is from B ( 0, 5 ). Write an equation that expresses all points that satisfy that condition.

22 )1()2( ++− yx = 22 )5()0( −+− yx → x2 – 4x + 4 +y2 + 2y + 1 = x2 + y2 – 10y + 25 → simplify → ........ _____________ 5. Show whether the following functions are even, odd, or neither

a) f(x) = 3 b) f(x) = 2 2x

f(-x) = 3 , since f(x) = 3 for any value of x f(-x) = 2 , no change → f(x) is even 2x

including –x. → f(x) must be even. Quiz # 6, Feb. 11, 2003

1. Complete the following right triangles C B A C B A side a = 24, side c = 24 2 side a = 12, angle B = 600 find angle B→ ________ and side b _______ find side b = ________ and side c. ________ ans: angle B = 45o, side b = 24 ans. side b = 12 3 , side c = 24 2. True or False a)The following represents the graph of y = sin x . → __________ b) the following represents the graph of y = 2x + 4 → ____________ ans. No – you need intercepts, periods, ans. No, you need intercepts 3. x2 - 4 = 0 can only be zero when x = 2. True or False. → _________________ ans. false → x = 2 ± 4. To find the length of a diagonal of a given rectangle, you use ? _______________________ ans. distances To compare that two diagonals are of equal length you would ? ans. find their lengths by the use of distances and compare them 5. If you are trying to prove that two segments are perpendicular, then you can use the fact that the product of their slopes = -1. True or False. → ________________ ans. no → you prove that statement , you do not use it 6. The easiest and quickest way to determine if three points are collinear is to → slopes , distances work, but slopes is a little bit quicker 7. If P is a point that is 12 units away from point A and P happens to 4 units away from point B with P on the segment AB extended through B, then what is the value of r ( in the formula for the division of a line segment). | P x = x1 + r(x2-x1) r = ________ | B A | ans. r = AP / AB = 12 / 8 = 3/2 Quiz # 7, Feb. 13, 2004

1. The domain of the function y = Arcsin x is all x/ such that -1 ≤ x ≤ 1 while the range is -π/2 ≤ y ≤ π/2 What are the domain _________________ and the range _________________ of the function y = Arccos x. ans. domain: - 1≤ x ≤ 1 ans. 0 ≤ y ≤ π 2. If f(x) = 2x, and g(x) = ( x – 2 ) 2, then find f og (1) = f(g(1)). ____________ ans. g(1) = 1 → f( g(1) ) = f( 1) = 21 = 2 3. If f(x) = 2x – 4 and g(x) = (x+4)/2, find g o f( 13 ). ____________ ans. f(13) = 22 → g (f( 13)) = g( 22) = (22 + 4)/2 = 13, Do you know what this says about g and f ? 4. Find the slope of each of the following lines a) a line that is perpendicular to the line passing through the segment AB, A(3, -2) and B(-3, 4) . → __________ ans. slope of AB = ( 4 +2) / ( -3 – 3 ) = 6 / -6 = - 1 b) a line that is parallel to the line x = 3 . → _______________ ans. slope = m = undefined c) x/2 - y/3 = 1 → _______________ ans. –y/3 = -x/2 + 1 → y = 3x/2 - 3 → slope = 3/2 5. What is the equation of the line that a) has slope 2 and crosses the y-axis at the point ( 0, - 4 ) → _________________ ans. y = 2x – 4 b) has no slope ( undefined slope) and passes through the point ( 2, 5) → ____________ ans. vertical line → x = 2 c) is parallel to the line x + 2y = 3 and passes through the origin. → _____________ ans. x + 2y = 0 d) is the perpendicular bisector of the segment (2, -1) and (4, 1) → ____________ ans. slope = -2/2 = -1 point ( 4, 1 ) y = mx + b → y = -x + b → 1 = - 4 + b → b = 5 → y = - x + 5 Name _________________________________________ Math 1321 – Quiz #8 , Feb. 16, 2004

1. Find the solution of the following system of equations by using the elimination method. ax + y = c x – by = d Solve for x only. x = ___________ ans. ax + by = bc x – by = d ---------------------------- ax + x = bc + d → (a + 1) x = (bc + d) → x = (bc + d) / ( a + 1 ) 2. Use the substitution method to solve for y ( solve only for x) x2 - y2 = 4 ( hyperbola ) x – y = 3 ( line ) ans. x = y + 3 → ( y + 3) 2 – y2 = 4 → y2 + 6y + 9 – y2 = 4 → 6y = -5 → y = - 5/6 3. Find the directed distance from the line 4x – 3y = 6 to the point ( -1, 0 )

ans. d = 22 )3(4

6)0(3)1(4

−+−

−−− = -10 / - 5 = 2

4. What is the diameter of a circle with center at the point ( -1, 0) and tangent to the line 4x – 3y = 6 ans. distance formula should give you the radius , double it to get the diameter

r = 22 )3(4

6)0(3)1(4

−+−

−−− = -10 / - 5 = 2 → diameter = 2r = 4.

5. Two pipelines are parallel to each other. If they are 4 meters apart and the first pipeline can be represented by the equation 12x + 5y = 12 (HINT: the line you are looking for is of the form 12x + 5y + K = 0 and is 4 units away from the line 12x + 5y = 12 ) there are two possible answers – depending on whether you are trying to find the line above or below ans. Decide which line you are going to find – I choose the one below. Find a point on one of the lines – you really can not find a point on the line you are looking for – you do not even know what the equation looks like. Find a point on 12x + 5y = 12 → an obvious point ( 1, 0 ) Use P( 1, 0 ) and the line 12x + 5y + k = 0 ( remember this line is below the original one → so, distance d = 4 Why ? Why not - 4 ? Use the distance formula → set it equal to 4 → you are looking for k and then the line. Name ________________________________________________ Math 1321 – Quiz #9 – Feb. 18, 2004

1. Describe each of the following curves by giving a complete description. Include things such as slope, center, radius, or the exact description(1d). a) y = - 2 → _____________________________________________________________________ ans. horizontal line passing through (0, -2 ) b) y = x/3 – 6 → ________________________________________________________________ ans.: slant line, m= 1/3 , y-intercept = - 6 c) (x – 2 ) 2 + ( y – 4 )2 = 9 → ________________________________________________________ ans.: circle with center at (2, 4) and radius 3 d) x2 + y2 = 0 → _______________________________________________ ans.: point, (0, 0 ) y-axis 2. Graph the following curve. (x+2)2 + y2 = 9 / / / / / / / x-axis ans. Circle with center at (-2, 0), r = 3 → x-intercepts= -1 and -5 Domain: all x’s, -5 ≤ x ≤ 1 Range: all y’s , -3 ≤ y ≤ 3 3. What is the equation of the circle with center at (2. -5 ) and radius = 7. __________________________________ ans. ( x – 2)2 + (y + 5)2 = 49 4. Why is ( x + 2)2 + ( y -1)2 = - 4 not a circle ? __________________________________________________ ans.: “radius=r2 = - 4 <0” not possible with sum of two nonnegative real numbers. 5. What is the equation of the circle with center at (2, 3) and passing through the point (-1, - 1 ) ? ____________________

ans. ( x – 2)2 + (y -3)2 = ( 22 )13()12( +++ )2 = 25 ( x -2)2 + (y-3)2 = 25 Quiz #10 – Feb. 20, 2004

1. Write the following in the general form of a circle. (x +2)2 + (y-2)2 = 4 ans. square and combine similar terms, set = 0 x2 + 4x + y2 -4y + 8 = 4 → x2 + y2 + 4x – 8y + 4 = 0 2. Write in the standard form of a circle. x2 + 4x + y2 – 2y – 4 = 0 ans. complete the square properly and write in the form (x –h)2 + (y-k)2 = r2 x2 + 4x + 4 + y2 – 2y + 1 = 4 + 4 + 1 → (x + 2)2 + ( y -1)2 = 9 3. What is the equation of the circle with a) center at ( 3, -2 ) and tangent to the line 3x – 4y = 2. ans.: find the distansce from the line 3x – 4y -2 = 0 to the point (3, -2 ) and remember that the radius is always positive

22 43

2)2(4)3(3

+−

−−− = 15/( - 5) = - 3 → radius = 3 → (x – 3)2 + (y+2)2 = 9

b) If the circle has its center on the x-axis and also on the line 4x – y = 8 and has a diameter of length 12, then find the equation. ans.: Center is at the point (x, 0) and is also on the line 4x-y = 8 → the point (x,0) is on the line.

This allows you to solve for x. Now you have the center and the diameter, which means you have the radius. If y = 0 , then 4x – (0) = 8 → x = 2 → center C ( 2, 0) and diameter = 12 → r = 6

→ (x -2)2 + y2 = 36

Qz #11 – Feb. 23, 2004

1. If f , f(x) , represents a constant function, then f(x) must be nondecreasing and nonincreasing. Pf. Begin with function f being a constant function, f(x) = c. Prove that it is nondecreasing and nonincreasing. Let x1 and x2 be given with x1 < x2. a) If f is increasing, then f(x1) < f(x2), but f(x1) = c and f(x2) = c. So, f(x1) is not < f(x2) which means f can not be increasing → f is nonincreasing. Also, b) If f is decreasing, then f(x1) > f(x2), but we know that f(x1) = c and f(x2) = c f decreasing means that f(x1) > f(x2), but in this case we have c is not > c → f is not decreasing We conclude that f is nondecreasing, nonincreasing. 2. Suppose that a pipe leans against a wall and a cross section has the appearance below. Find the equation of the circle represented by the cross section. The circle is tangent to both sets of axes and is of radius 4. The line L has slope crosses the y-axis at y=12 and the x axis at x = 16. C: __________________________ ans. You are given the answer → center is at (4, 4) because of the tangency and you know the radius to be 4 → (x-4)2 +(y-4)2 = 16 Remember where this problem came from (HW ) look over it – there are different ways to look at it –different

questions 3. Find an angle created by the intersection by the intersection of the lines φ = ______________ (Short and Sweet) Should not need more space than what is provided 4x – 3y = 4 3x + 4y = 1 These lines happen to be perpendicular. Why ? Since they are → φ = 90o 4. Find the acute angle bisector created by the intersection of the lines equation1: 3x – 4y = 3 d1 > 0 and d2 < 0 so equation2: 4x – 3y = 4 2 d1 ≠ d2 but d1 = - d2 1 x – y – 1 = 0 Pick a point on the line that bisects the acute angle, say P(x1, y1) - later change to (x,y). Compare distances d1 and d2 d1 ≠ d2 but d1 = - d2 where d1 is the distance from line 1 to P and d2 is the distance from line 2 to P. Name ___________________________________________ Math 1321 – Quiz #13, Feb. 27, 2004

1. Write down the translation formulas when the origin is translated to the point O/ (2, -1 )

( there should be no h or k in your formla) ans. x = x/ + h , y = y/ + k or x/ = x –h , y /= y - k 2. Find the new coordinates of P(2, -4) when the origin is moved to O/ ( -2, 3) . __________________ ans. x/ = x – h = 2 – (-2) = 4 and y/ = -4 – (3) = - 7 → P/ ( 4, - 7) What will the equation x + 2y = 3 look like in terms of x/ and y/ after the translation ? __________________________________ solution: x= x/ + h = x/ + (-2) and y = =y/ + (3) x + 2y = 3 → ( x/ + (-2)) + 2(y/ + 3 ) = 3 → x/ + 2y/ - 2 + 6 = 7 → x/ + 2y/ = 3 3. Find the point to which the origin must be translated to so that the new equation ( in terms of x/ and y/) a) (x -2)2 + ( y+5)2 = 9 will be free of all first degree terms, what does the equation look like ? point: ______________ equation: _____________________ ans. (2,- 5) ans. (x/ )2 + (y/ )2 = 9 b) Same as above with the hyperbola (x -2)2 - ( y+ 3)2 = 4 point: _______________ equation: _____________________ ans. (2, -3) ans. (x/ )2 - (y/ )2 = 4 4. Find the point to which the origin must be translated to so that the new equation will be free of one first degree term

(just one) and the constant term. x2 - 4x = 2y - 6 point: _________________ equation: ____________ ans. You can replace x with x = x/ + h and y with y = y/ + k isolate the x/ term and the constant term → set equal to zero and solve for h and k. Done! or Complete the square: x2 - 4x + 4 = 2y – 6 + 4 → ( x – 2)2 = 2y – 2 → (x-2)2 = 2( y – 1) → point: (2, 1) → equation: (x/ )2 = 2y/ Name ______________________________________________ Math 1321 – Quiz #14, March 1,

1. Identify each of the following by using the term that best describes the graph of the equation point (what point), vertical line, horizontal line, slant line, circle, point, no graph,

a parabola that opens to the right-left-up-down a) (x+2)2 + (y-3) = - 1 → __________________________________ ans.: a parabola , opens downward that problem should have read → (x+2)2 + (y-3)2 = -1 and in that case the answer would have been→ no graph b) y = - 2 → _____________________ ans.: horizontal line c) y2 = ( x + 2 ) → ________________ ans.: parabola that opens to the right d) x2 + y2 + 2x + 4y - 400 = 0 → ____________________ ans.: circle e) x2 – 2x + y - 1 = 0 → _______________________ ans.: parabola that opens downward f) ( x+2 )2 + ( y – 1 )2 = 1 → ___________________________ ans.: circle with radius 1 2. Find the domain of a) ( x +1)2 + y2 = 1 → _________________________________________ ans.: -2 ≤ x ≤ 0 b) y2 = x + 1 → ___________________________________________ ans.: x ≥ - 1 c) x = 3 → _________________________________ ans.: { 3 } 3. Find the range of a) y2 = x+ 1 → _____________________________________ ans.: all real numbers b) x2 + (y-1)2 = 1 → ________________________________ ans. 0 ≤ y ≤ 2 4. Find the equation of the parabola with a) vertex at the origin and focus at F(-2, 0 ). → _________________________________ ans.: y2 = - 8x b) directrix y = -3 and Focus at F(0, 3) → _________________________________ ans.: x2 = 12y Name ______________________________________ Math 1321 – Quiz #15, March 3, 2004

1. Write down the general form of a parabola that opens to the sides ( left, right)

ans.: y2 + Dx + Ey + F = 0 2. Write down the standard form of a parabola that opens up or down. ans.: ( x – h)2 = 4a(y-k) 3. What is the equation of the parabola with a) focus at ( 2, -3) and vertex at V( ( 0, - 3 ) ans.: a = + 2 and the equation is of the form ( y – k)2 = 4a(x-h) (h,k) = ( 0, -3) ( y – (-3))2 = 8 (x – 0 ) → ( y + 3)2 = 8x b) l.r. of length 12, opens down with vertex at ( 2, - 5) ans.: since we know the length of l.r. is 12 → 4a = ± 12 (depending which way it opens) - in this case (negative) opens down → ( x – h) = 4a(y-k) and vertex (2, -5) = (h, k ) 2

→ ( x – 2)2 = - 12 (y → 5) c) passes through the point ( 2, 0 ) and axis of symmetry parallel to the x-axis with vertex V(1, 4) ans.: Axis of symmetry parallel to x-axis: ( y- k)2 = 4a (x – h) with V(1, 4) = (h, k) → ( y – 4)2 = 4a(x – 1) since we know a point (2, 0) replace x = 2 and y = 0 solve for 4a → ( 0 – 4)2 = 4a(2-1) → 16 = 4a (1) → 4a = 16 → solution: (y – 4)2 = 16(x – 1) 4. Symmetry: a) a curve is said to be symmetric about the x-axis if (x,y) is a point on the curve, then so is the point ans.: ( x, - y) ____________ b) If for every point P(x,y) on a given curve the point Q( -x, - y) is also on the curve, then what type of symmetry does this curve have ? ans.: symmetry about the origin Math 1321- Short Quiz -#16 – March 8, 2004

1. Write the standard form of an ellipse with center at (h,k) and minor axis parallel to y-axis.

ans. 1)()(2

2

2

2

=−

+−

bky

ahx

2. Write the standard form of a hyperbola with transverse axis (major) parallel to y-axis and center at (h,k)

ans. 1)()(2

2

2

2

=−

−−

bhx

aky

3. Find the equation of an ellipse whose sum of the distances from two fixed points is 60 and endpoints of minor axis are at a) B(3, -2 ) and B/ ( - 7, - 2) → sum = 2a = 60 → a = 30 → b = 10/2 = 5

ans.: 15

)2(30

)2(2

2

2

2

=+

++ yx

b) we defined b2 = ___________________ ( for an ellipse ) ans.: b2 = a2 - c2

c) what is the eccentricity of this ellipse ? e = c/a = _______________ ans.: c/a = 30875

c2 = a2 - b2 = 900 – 25 = 875

4. The equation of a hyperbola is given by 114

)2( 22

=−+ yx

. Find

a) length of the conjugate (minor) axis . _____________ ans.: b = 1 → 2b = 2 b) length of the l.r. ( 2b2/a ) . _______________ ans.: 2b2/a = 2(1)/2 = 1 c) we defined b2 = _______________ ans.: b2 = c2 - a2 e) asymptotes: in the form ax + by = c

ans.: 012

)2(=−

+ yx 0

12)2(

=++ yx

multiply by 2 to simplify

x + 2 – 2y = 0 → x – 2y = - 2 and x + 2y = - 2 5. A satellite travels in an elliptical path around a planet. If the farthest the satellite gets from the planet is 180 units and the closest it gets is 20 units then find an equation that represents the orbit of the satellite. Assume that the planet is a fixed point. Draw a picture: distance from a focus to a vertex → closest: a – c = 20, farthest: a + c = 180 → 2a = 200 → a = 100 c = 80 . So, b = 1002 – 802 = 10000 – 6400 = 3600

160100 2

2

2

2

=+yx

Name ________________________________________________ Math 1321 – Quiz #17, March 24, 2004

1. We finished studying conics. List the three conics we studied ( circles could be included – but the other three would be ?)

Answers: ellipses, parabolas, hyperbolas 2. Find the x-intercepts of a) f(x) = x2 – 2x → _________________ Let y = 0 → x2 – 2x = 0 → x(x – 2) = 0 → x = 0 and y = 2

b) y = 22

−+

xx

→ ____________________ let y = 0 → 0 = 22

−+

xx

but that means → 0 = x + 2 → x = - 2

3. What are the y-intercepts of a) f(x) = (x+2) (x – 3 ) → _________________ let x = 0 → f(0) = ( 0 + 2 ) ( 0 – 3 ) → y = 2(-3) = - 6 b) y = - 2 → _______________ this is a horizontal line that crosses the y-axis at y = -2, :. y-int: y = -2

c) y = 22

−+

xx

→ ____________________ let x = 0 → y = 2020

−+

= 2/(-2) = -1 → y-int = - 1

4. What are the vertical asymptotes of

a) y = 22

−+

xx

→ ____________________ if it does not factor, set denominator equal to zero → x -2 = 0

x = 2 ( vertical line: vertical asymptote)

b) y = 92 −x

x → ________________ set x2 – 9 = 0 → x = 3 and x = - 3 we get two vertical asymptotes

5. What are the horizontal asymptotes of

a) y = 222−+

xx

→ ____________________ Since degree on top is equal to degree in denominator → y = x

x2

= 1/2

H. A. : y = 1/2

b) y = 92 −x

x → _________________ Since degree in denominator exceeds degree in numerator → y =0 is the H.A.

Name ______________________________________________________ Math 1321 – quiz #18, March 26, 2004

1. Sketch the graph of y = x – 4 ans.: line that crosses the x-axis at x=4 and the y-axis at y = - 4.

2. Sketch the graph of f(x) = 4162

+−

xx

x-intercept: _________ y-intercept: ____________ V.A. ___________ H.A. _________

ans.: factor first then find each of the values above. → f(x) = 44

)4)(4(−=

++− x

xxx

if x ≠ - 4

you know the graph of y = x -4 ( answer from #1 above) except that it has a missing point ( a hole) at x = - 4. It has no asymptotes

3. Sketch the graph of y = xx

xx2

322

2

+−−

x-intercept: _________ y-intercept: ____________ V.A. ___________ H.A. _________

y = )2(

)1)(3(++−

xxxx

x-int: (x – 3) (x + 1) = 0 → x = 3 or x= -1

y-int: y = )0(20

3)0(202

2

+−−

= 03−

not one

V. A.: x(x+2) = 0 → at x = 0 and at x = - 2 ( lines ) H.A.: y = 1/1 = 1 Name ___________________________________ Math 1321 – Quiz #19, 2004

1. Find the vertical, horizontal, and vertical asymptote(s) of

b) y = 4

342

−+−

xxx

V. A. : ___________ H.A. : ___________________ S.A.: __________

y = 4

)1)(3(−

−−x

xx → V. A.: (x – 4) = 0 → x = 4 (line )

since the degree in numerator is larger than degree in denominator, there is no H.A.: S. A. : Since there is no canceling and the degree in numerator exceeds denominator by one: there is one

( x 2 – 4x + 3 ) ÷ ( x – 4 ) = x + 4

3−x

( either by long-hand division or synthetic division )

→ y = x is the slant asymptote b) xy + y – x2 + 1 = 0 → y = V. A. : _________________ H.A. : ___________________ S.A.: ______________

xy + y – x2 + 1 = 0 → y(x + 1) = x2 – 1 → y = 112

+−

xx

= )1(

)1)(1(+

+−x

xx= x - 1 → y = x- 1 with x ≠ - 1

this is a line with a missing point – no asymptotes

2. Given y = 1

432

2

+−−

xxx

, we find that y = 1 is a horizontal asymptote.

Does the the curve cross it- if so, at what point P(x,y). Show algebraically-do not need a graph. solution: if y = 1 is the horizontal asymptote → y is replaced with 1, we can find ( maybe ) a value of x on the curve ( otherwise: it does not cross it )

1 = 1

432

2

+−−

xxx

→ x2 + 1 = x2 - 3x - 4 → 1 = - 3x – 4 → - 5/3 = x. The curve crosses the H.A. at

x = -5/3

3.. Given y = 2

322

−−+

xxx

we get y = 2

322

−−+

xxx

= x + 4 - 2

11−x

What is the slant asymptote ? _____________ y = x + 4

Does the curve cross it ? replace y with y = x +4 and solve for x. →y = 2

322

−−+

xxx

x + 4 = 2

322

−−+

xxx

→ x2 + 2x – 8 = x2 + 2x – 3 → - 8 = - 3 false :. no intersecting point

Name _______________________________________________ Math 1321 – Quiz #20, April 2, 2004

1. Define y = cosh x ( not in words but in terms of its mathematical definition – equation )

cosh x = 2

xx ee −+

2. Sketch the graph of y = sinh x. You need to draw the asymptotes to this function and label them. Solution:

sinh x = 2

xx ee −−, draw y = ex and y = - e-x

and use these graphs to sketch sinh x. 3. Find 2sinhx • cosh x by using their definitions and simplifying ( with trig functions- 2sinx cos x = sin 2x.

ans.: 2sinh x • cosh x = 2 (2

xx ee −−) •(

2

xx ee −+ ) =

2

22 xx ee −− = sinh 2x

4. Find x a) x = - 24 → x = ____________ b) log ½ 8 = x → x = ____________ ans.: x = - 16 ans.: (1/2) x = 8 → x = - 3 c) If log

3 y = 0.24, then what does log 3 y5 = x ? x = ____________ ans.: log 3 y5 = 5 log 3 y = 5 ( 0.24 ) = 1.20 5. What is the domain of y = log 4 x ? ________________________________________ domain: x > 0 6. What is the range of y = - 3 x

→ __________________________ range : y < 0

7. If f(x) = 2x – 6 and g(x) = 2

6+x and you know that f and g are inverses of each other, then what does

f o g (r) = ? _________ ans.: anytime that f and g are inverses, we get f o g (x) = f ( g(x) ) = x → f o g ( r ) = r. ( as long as f and g are defined at x = r ) Name _______________________________________ Math 1321 – Short Quiz # 21 (?) - April 5, 2004

1. Graph the points ( polar coordinates ) a) P ( 3, - 200o ) θ=90o b) Q( -2, - π/4 ) θ=180o polar axis θ = 270o P should be three units from the pole with and angle of θ = - 200o ( “quadrant II” ) Q should be two units from the pole with θ = - π/4 (“quadrant II” ) 2. Write down the formulas to change from rectangular to polar coordinates tan θ = ____________ tan θ = y/x r2 = _______________ r2 = x2 + y2 3. Write down the formulas to change from polar to rectangular x = ____________ y = __________ x=rcos θ y = rsin θ 4. Change to polar coordinates a) ( 3, - 3 ) → ______________ with | θ | < 360o r2 = 32 + (-3)2 = 18 → r = 23± and tan θ = y/x = -3/3 = -1 → θ = - 45o .

(3, -3) is in quadrant II -- use P(- 3 2 , - 45o ) b) ( )34,4 → _____________ with | θ | < 360o

r2 = 42 + ( 4 3 )2 = 16 + 48 = 64 → r = 8± , and tan θ = 4

34= 3 → θ = 60o

So, ( 8, 60o ) c) ( 0, - 2 ) → ______________ with | θ | < 2π → r represents the distance from the origin to P, so r = 2 with θ = - π/2 → ( 2, - π/2) 5. Change to rectangular coordinates ( exact values ) a) ( 4, - 60o ) → x = _______ y = ________ x = rcos θ = 4 cos ( - 60o ) = 4 ( ½) = 2, y = rsin θ = 4 sin ( -60o ) = - 2 3 → ( 2, -2 3 b) ( -4, - 3π/2) → x = _______ y = ________ x = rcos θ = - 4 cos ( -3π/2) = 0 , y = rsin θ = -4 sin ( - 3π/2) = -4 → ( 0, - 4 ) Name ______________________________________ Math 1321 – Quiz 22– April 14, 2004

1. In the equation y = mx + 2, how many different complete curves ( more than just a point ) can be generated by letting m=2 You only get one curve – one curve for each value of m that you select In the equation x = 2t2 , y = t, how many different curves(more than just a point) can be generated by letting t = 1. you get only one point for each value of t --- select all permissible values of t, you will get one complete curve. 2. What kind of curve does each of the following polar equations represent a) r = 4 sin 4 θ → ____________________ rose curve ( amplitude 4) with 8 petals b) r2 = 9 cos 2 θ → ______________________ lemniscate with primary axis at θ = 0 o c) r = 5 - 3 sin θ → ______________________ limacon → heavy curve (“fat circle”) around the origin 3. There are three types of limacons → curve around the origin, curve with an inner loop, what is the other one ? cardioid 4. How many petals does the rose curve r = 4 cos 4θ → 8 petals 5. Give me quick graph of r2 = 4 sin 2θ → lemniscate with primary axis at θ = 45o and amplitude 2 Name _____________________________________ Math 1321 – Quiz # 23, April 16, 2004

1. Give me a parametric equation of each of these two a) 2x – 3y = 6 → b) y = x2 + 2x – 4 → ______________________________ ___________________________ let y = t, then x = (3t + 6 ) / 2 let x = t, then y = t2 + 2t =- 4 2. Eliminate the parameter in the equation(s) and write the equation in terms of x and y a) x = 2t - 2 → _________________________ t = y – 1 → x = 2( y – 1 ) – 2 → x = 2y – 2 – 2 y = t + 1 So, x – 2y = - 4

b) x = 2+t

t, y =

23+tt

→ ____________________ y = 2

3+tt

= 3 •2+t

t= 3 ( x ) → y = 3x

c) x = 2t, y = 4t → _______________________ Now, y = 4t = (22 ) t = ( 2t )2 → y = x2 ( they may not represent the exact same graph ) d) x = sinh t, y = cosh t ( HINT: cosh2 x - sinh 2 x = ? ) → _____________________ Recall that cosh2 x – sinh2 x = 1

→ x2 = sinh 2 t and y2 = cosh 2 t → y2 - x2 = 1 3. Sketch the actual graph of the given parametric equations. a) x = sin2 t, y = cos 2 t b) x = sin2 t , y = cos t x + y = 1, which is a line – but x + y2 = 1 ( since y2 = cos 2 t and sin2 t + cos2 t = 1 ) 0 ≤ sin2 t ≤ 1 this is a parabola but we only use the portion in which 0 ≤ cos2 t ≤ 1 0 ≤ sin2 t ≤ 1 ( 0 ≤ x ≤ 1 ) and - 1≤ cos t ≤ 1 ( 0≤ y ≤ 1) Draw a line segment with 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1. section of a parabola Name __________________________________ Math 1321 – Quiz #24 --- April 21, 2004

1. Plot the points A(3, 4, 0 ) and B( 4, 7, 9 ) z - 9 • B(4, 7, 9) – more or less y

/4 / 7 x 3-------------• A(3,4, 0) 4 ---------------| 2. Find the midpoint of A and B above. midpoint: x = (3+4 )/2 = 7/2, y = (4 + 7)/2 = 11/2, z = (0+9)/2 = 9/2 → ( 7/2, 11/2, 9/2 ) 3. Find the distance between A and B above.

distance = d = 222 )90()74()43( −+−+− = 8191 ++ = 91 4. Find a point P that is twice as far from A as from B and is between A and B ( along the segment AB). r

r = BAPAr = 2/3,

x = x1 + r( x2 – x1 ) = 3 + 2/3( 4 – 3 ) = 3 + 2/3 = 3 2/3 or 11/3 y = ........................ = 4 + 2/3 ( 7 – 4 ) = 4 + 2 = 6 z = ........................ = 0 + 2/3 ( 9 – 0 ) = 6 → ( 11/3, 6, 6 ) 5. Sketch the graph of a) x + 2y + 3z = 6 b) x + 2z = 4 --2 -- 2 | 3 6 -- 4 --

Name __________________________________________ Math 1321 – Quiz #26 -- April 26, 2004 1. Draw and label a vector --- tail, head head tail 2. Use the two given vectors to find the resultant vector ( A + B ) . 3. Construct a parallelogram by using the two given vectors in their current position to describe (draw) the difference of A and B ( A – B). A-B A B 4. Find a vector of length 1 having opposite direction of the positive x-axis. ________________ ans.: i has length 1 → - i has length 1 and is in the opposite direction of the positive x-axis 5. Find any vector that has length 8. ___________________ ans.: 8i has length 8, so does 8j --- there are others, but these are the easiest to think of 6. What do we call a vector of length 1 ? _____________________ ans.: a unit vector 7. What is the length of the vector -3i - 4j ? | -3i - 4j | = ______________

ans.: by definition | -3i - 4j | = 22 43 + = 25 = 5 Name _______________________________ Math 1321 – Quiz # 27 – April 28, 2004

1. Let vectors A and B be points on the plane with A = ( 2, 4 ) and B = (- 3, 4 ) ( write your answers in the form ai + bj ) - Find a vector

from the origin to point A from the origin to point B → __________ → ___________ answer: 2i + 4j answer: -3i + 4j A and B below represent the vectors from the origin to the points A and B, respectively a) Find A + B = ______________ answer: - i + 8j b) Find A – B = ______________ answer: 5i + 0j c) Find 3A = _________________ answer: 6i + 12j

d) Find a vector of length 1 having the same direction as vector B. answer: B / | B | = jiji54

53

543

+−=+−

e) Find a vector from A to B. answer: A to B = B – A = - ( A – B ) = - ( 5i + 0j) = - 5j 2. Given that vectors A and B are parallel but not equal.

Relate A and B in terms of an equation. → answer: A=tB or B = sA 3. If P is the midpoint of the segment AB, and OA and OB represent vectors from the origin to A and B respectively, then Write an equation for OP in terms of the other vectors. OP = ____________________ answer: OP = A + ½ ( B – A ) 4. In class we defined the dot product (scalar product ) by A • B = | A | | B | cos θ, where 0 ≤ θ ≤ 180o If A • B = 0 and A and B have nonzero lengths, then what would θ equal ? ____________ What does that say about the relationship between A and B ? answer: A • B = 0 → 0 = | A | | B | cos θ, but neither A nor B = 0 → cos θ = 0 → θ = 90o → A and B must be perpendicular Name _______________________________ Math 1321 – Quiz # 28---- May 5, 2004

1. Find the dot product of

A= 3i – 2j + k and B = i – 4j + 2k . → __________

answer: 3(1) + (-2)(-4)+ (1)(2) = 3 + 8 + 2 = 13 Is A • B = B • A ( commutative )? __________ answer: yes 2. If A = < 2, 1, 1 > and B = < 1, 2, c > what does c have to equal so that A and B represent perpendicular vectors. c = _________ answer: 2(1) + 1(2) + 1(c) = 0 → c = - 4 3. What is the angle θ between vectors A and B ( from # 1 ) ? ______________________

answer: A • B = | A | | B | cos θ → cos θ = |||| BA

BA • =

211413×

= cos θ, use calculator to find θ

4. If A = 2i – 3j + k, find a vector B that twice as large as A and has the same direction.

answer: B = 2A = 4i – 6j + 2k 5. Find the equation of the plane that is perpendicular to the vector N = 2i – 3j + 5k and passes through the point (1, 1, 0 ) . → ___________________________ answer: 2x – 3y + 5z + d = 0 → 2x -3y+5z -1 = 0 6. Find one vector that is perpendicular to the plane 2x – y + 5k = 1 . → _________________________ answer: N = < 2, -1, 5 > = 2i – j + 5k 7. What is the equation of the plane that is parallel to the plane 2x + 3y + 3z = 2 and passes through the origin ? → ________________ answer: 2x + 3y + 3z = 0 8. What is the determinant of

→ → det = [ -12 + 0 – 16 ] - [ 0 - 12 – 4 ] = -12 ( maybe )

−−−

−

440332

211

431

02

1

440332

211 −−

−−−

−

Name __________________________________ Math 1321 – Quiz --- May 7, 2004

1) If C = A • B, what can you say about C ? C is what type of quantity ? _____________ C is a scalar 2) mA ? What type of quantity ? ____________________________ mA is a vector 3) A x B = B x A ? __________________ not equal 4 ) If C = A x B, then what can you say about C ? a) C is what kind of quantity ? _________________________________ vector b) magnitude: __________________ | A x B | = |A| |B| sin θ c) direction: ________________________ right hand rule 5) What is the equation of a line that a) is parallel to 3i + 4j – k and passes through the point ( 3, 1, 1 ) ? x = 3t + 3 , y = 4t + 1, z = -t + 1 b) passes through the points A( 4, 0, 2) and B( -1, 2, 1 ) ? V = (4 – (-1) ) i + ( 0 – 2 ) j + ( 2 – 1 ) k x = 5t + 4 , y = -2t + 0 , z = k + 2 6. Find A x B if A = 2i – j – k and B = 4i + 2j + k . As done in class: