Name: _____________________ Practice Test Chapter 5

NO CALCULATORS

Find each of the following integrals. Show all substitutions used(when necessary).

1. dxxx22

)3( +∫ 2. dxxx∫

− sin

12

3. ∫ − dxxx 13632

4. dxxx )6sin()6cos(∫

Evaluate the following. Show all work.

5. ∫ +4

0

2 )sec4(

π

dxxx 6. ∫ −

3

2

22)3(

dxx

x

7. Suppose that f and g are continuous functions and that

13)(

5

2

=∫ dxxf , 9)(

5

2

=∫ dxxg , 5)(

2

0

−=∫ dxxf , and 20)(

8

2

=∫ dxxf

Find each of the following:

a. dxxf∫5

0

)( b. [ ]dxxgxf∫ −

5

2

)()(3 c. dxxf∫5

8

)( d. [ ]dxxf∫ −

5

2

)4)(3

8. Given that 13 += xdx

dy. Find the equation for y if 7)2( =y .

9. The acceleration of a particle is given by the function a(t) = 2t - 3 m/s2. Given v(0) = 2

a) Find the displacement of the particle on the interval t = [0, 2]

b) Set up but do not solve an integral expression to find the total distance travelled by the particle on

the interval t = [0, 2]. Your integral expression may NOT contain an absolute value.

10. Find a value x* on [0, 3] which satisfies the Mean Value Theorem for 24)( xxxf −=

11. Find the average value of the function 1)( 3 +−= xxxf on the interval from [0, 2].

12. Given xxf 2)( =′′ , 1)2( −=′f , and 1)3( =f

Find f(x)

CALCULATORS ALLOWED

13. Find the area under the curve xy 2= from x = 0 to x = 4 using the following approximations. For

each be sure to show your graph, set-up, and intermediate calculations!! Let n = 4.

a) Right endpoint rectangles. b) Trapezoids

14. Estimate ∫90

0

)(xf by using a trapezoidal approximation with 6 equal subdivisions.

x 0 15 30 45 60 75 90

f(x) 12 11 10 10 8 7 0

15. Evaluate dxxx∫5

2

)2)(cosln(

16. Draw a slope field at the specified points for the differential equation xdx

dy 1=

Sketch a possible solution going through (1, 1).

17a. Find dttdx

dx

∫−

+1

31 17b. Find ∫ +

4

2 5x

t

dt

dx

d

18. Given the graph of )(tg shown at the right

Given dttgxf

x

∫=0

)()( , and 1)0( =f , find each of the following:

a) )3(f

b) )2(−f

c) )2(f ′

d) )1(−′′f

e) Interval(s) where )(xf is decreasing

f) Value(s) of x where )(xf has a relative maximum or minimum. Justify your answer.

g) Interval(s) where )(xf is concave up

h) Value(s) of x where )(xf has an inflection point. Justify your answer.

)(tg