FE Review { Calculus, Di erential Equations, & Numerical...
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FE Review – Calculus, Differential Equations, & Numerical Analysis Brody Dylan Johnson 1. Consider the circle given by x 2 + y 2 = 7. Find dy dx at the point ( √ 3, 2). 2. If f (x)= a -x , a> 0, what is f 0 (x)? 3. What are the maximum and minimum values of f (x)=2x 3 + x 2 - 20x + 4 on [-4, 2]? 4. Differentiate f (x)= x 2 x . 5. Evaluate lim x→ π 2 π - 2x cos x . 6. A pebble is dropped into still water, forming a circular ripple whose radius is expanding at a constant rate of 10 centimeters per second. When the radius reaches 20 centimeters, how fast is the area enclosed by the ripple increasing? 7. Where is f (x)=6x 2 - x 4 - 3 both increasing and concave down? 8. A cruise line offers a trip for $1000 per passenger. If at least 100 passengers sign up, the price is reduced for all the passengers by $5 for every additional passenger (beyond 100) who goes on the trip. The boat can accommodate up to 250 passengers. What number of passengers maximizes the cruise line’s total revenue? What price does each passenger pay in this case? Hint: You may assume the optimal number of passengers is at least 100. 9. Evaluate the definite integral R 3 0 2x 2 e x 3 dx. 10. Calculate R 2 0 1 - e -x dx. 11. Find the indefinite integral Z x +1 x(x - 1) dx. 12. Does the improper integral R ∞ 2 1 x(ln x) 2 dx converge or diverge? 13. A rod of length 2 meters and density δ(x)=3 - e -x kilograms per meter is placed on the x-axis with its ends at x = ±1. Find the center of mass, ¯ x. 14. Find the indefinite integral Z e 2x sin x dx. 15. Calculate the volume of rotation obtained by rotating the region between f (x)= √ x and g(x)= x about the x-axis for 0 ≤ x ≤ 1. 16. If f (x, y)= √ x 3 √ y find f x and f y . 17. Let f (x, y)=8+2x 2 - x +2y 2 - y. Find ∇f (x, y). 18. Find DivF and CurlF if F (x, y, z)=3x ~ i +(x - y) ~ j +(ze x +2y) ~ k. 19. A parametric curve is given by x = (2 - t) cos (2πt) and y = (2 - t) sin (2πt). Find the equation of the tangent line when t = 1 2 . 20. Find the general solution of the differential equation dy dx = y ln x.
Transcript of FE Review { Calculus, Di erential Equations, & Numerical...
FE Review – Calculus, Differential Equations, & Numerical AnalysisBrody Dylan Johnson
1. Consider the circle given by x2 + y2 = 7. Find dydx at the point (
√3, 2).
2. If f(x) = a−x, a > 0, what is f ′(x)?
3. What are the maximum and minimum values of f(x) = 2x3 + x2 − 20x+ 4 on [−4, 2]?
4. Differentiate f(x) = x2x
.
5. Evaluate limx→π
2
π − 2x
cosx.
6. A pebble is dropped into still water, forming a circular ripple whose radius is expanding at a constant rate of10 centimeters per second. When the radius reaches 20 centimeters, how fast is the area enclosed by the rippleincreasing?
7. Where is f(x) = 6x2 − x4 − 3 both increasing and concave down?
8. A cruise line offers a trip for $1000 per passenger. If at least 100 passengers sign up, the price is reducedfor all the passengers by $5 for every additional passenger (beyond 100) who goes on the trip. The boat canaccommodate up to 250 passengers. What number of passengers maximizes the cruise line’s total revenue?What price does each passenger pay in this case? Hint: You may assume the optimal number of passengers isat least 100.
9. Evaluate the definite integral∫ 3
02x2ex
3
dx.
10. Calculate∫ 2
01− e−x dx.
11. Find the indefinite integral
∫x+ 1
x(x− 1)dx.
12. Does the improper integral∫∞2
1x(ln x)2 dx converge or diverge?
13. A rod of length 2 meters and density δ(x) = 3− e−x kilograms per meter is placed on the x-axis with its endsat x = ±1. Find the center of mass, x̄.
14. Find the indefinite integral
∫e2x sinx dx.
15. Calculate the volume of rotation obtained by rotating the region between f(x) =√x and g(x) = x about the
x-axis for 0 ≤ x ≤ 1.
16. If f(x, y) =√x 3√y find fx and fy.
17. Let f(x, y) = 8 + 2x2 − x+ 2y2 − y. Find ∇f(x, y).
18. Find DivF and CurlF if F (x, y, z) = 3x~i+ (x− y)~j + (zex + 2y)~k.
19. A parametric curve is given by x = (2− t) cos (2πt) and y = (2− t) sin (2πt). Find the equation of the tangentline when t = 1
2 .
20. Find the general solution of the differential equation
dy
dx= y lnx.
21. Use Euler’s method to predict y(1) if ∆x = 0.5 and y satisfies
dy
dx= 2y3 + x2, y(0) = −0.5.
22. Solve the initial-value problem, d2ydx2 + 3 dy
dx − 4y = cosx, x(0) = 0, x′(0) = 1.
23. Find the Fourier series of the 2-periodic function which for 0 ≤ x ≤ 2 is given by
f(x) =
{1, 0 ≤ x < 1,
0, 1 ≤ x < 2.
24. Find the Laplace transform of f(t) = 1.
25. Use Laplace transforms to solve the differential equation,
dy
dt+ 4y = 2 + 3t, y(0) = 1.
26. Find the inverse Laplace transform of Y (s) =2s2 + 7s+ 32
(s+ 4)(s2 + 2s+ 10).
27. What is the approximation obtained by two iterations of the Bisection method for the root of f(x) = x2 − 2starting with L0 = 0 and R0 = 2? If an initial guess of x = 1 is used, what is the next estimate using Newton’sMethod?
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