Final Review Examples

1. Consider the function f(x) =1

x− 1when x > 1.

(a) Show that f is one-to-one.

(b) State the domain and range of f−1.

(c) Find an explicit formula for f−1. Does the domain and range from part (b) match the domainand range of the function you found?

2. The function f(x) = 7x−√x is one-to-one. Find the tangent line to the inverse function f−1(x)

at the point x = 6.

3. Simplify the expression

log5

(5x2+1

25x

).

4. Find dydx

for the following equations.

(a) (ln 2)(ln y) = 2x+y

(b) y = (tanx)lnx

5. A bacteria culture contains 300 cells initially and grows at a rate proportional to its size (growsexponentially). After 5 hours the population has increased to 600. When will the populationreach 9,000?

6. Express cos(cot−1(x)) as a function of x not involving trigonometric operations.

7. Evaluate the definite integral

∫ 1

0

arcsin(x)√1− x2

dx

8. Evaluate the limit

limx→0+

1

xx

9. Evaluate the integral or show that it is divergent∫ 2

−∞

1

x2 − 2x+ 2dx

10. Evaluate the integral

∫ 2

1

x3 ln x dx.

11. Evaluate ∫e2x sin(x)dx

12. Evaluate∫

sec4(θ) tan3(θ)dθ.

13. Evaluate: ∫x2√

1− x2dx.

14. Compute the integral∫

10(x−1)(x2+9)

dx.

15. Approximate the following integral with n = 6 using (a) Midpoint Rule and (b) Trapezoidal Rule.∫ 6

0

(x2 − 3) dx

16. Set up, but do not evaluate, an integral for the length of x = cos(y) on the interval 0 ≤ x ≤ 12.

17. Consider the following sequences. Determine if they converge or diverge. If one converges, statewhat it converge to.

(a){

(−1)nn3 + 2n− 1

4n3 + 1

}∞n=1

(b){n2 − 1

5n

}∞n=1

18. Find the sum of the following series:

∞∑n=0

3n + (−2)n

5n.

19. Find the sum of the following series:

∞∑n=4

[2n

n+ 4− 2(n+ 1)

n+ 5

].

20. Determine if the series converges absolutely, conditionally converges, or diverges.∞∑

n=14

(−1)n−1√n3 + n

21. Determine if the series converges absolutely, conditionally converges, or diverges.∞∑n=1

(−1)n(5n+ 1)

n

22. Determine if the series converges absolutely, conditionally converges, or diverges.∞∑n=1

(n2 + n

2n2 + 1

)n

23. The series∞∑n=1

(−1)n−14

n2is an alternating series which satisfies the conditions of the alternating

series test. Use the Alternating Series Estimation Theorem to determine the smallest k on the list

below so that the k-th partial sum is within1

100of the actual sum.

24. Find the center, radius of convergence, and interval of convergence of the following power series∞∑n=1

(−1)n−1(x− 2)n√n

.

25. Find a power series representation for the function

x2

(1− x3)2

in the interval (−1, 1). Hint: Differentiation of a power series may help.

26. (a) Find the Maclaurin series for f(x) = cos(x2)

(b) Express

∫ 17

0

cos(x2) dx as a series.

27. Find the smallest degree of a Taylor Polynomial about π that estimates sin

(13π

12

)with an error

less than .001. Write out the Taylor Polynomial.

28. Letdy

dx= y − x, y(0) = .5.

Use Euler’s Method with a step size of ∆x = .25 to estimate y(1).

29. Solve the following differential equation. Give the solution implicitly rather than explicitly.

dy

dx= xex

2−y

30. Solve the initial value problem:

dy

dx+ 2xy = x

y(0) = −4.

31. A tank initially contains 100 liters of salt water with 1 kilogram of dissolved salt. A well mixedsalt water solution containing 4 kilograms of salt per 100 liters is pumped into the tank at a rateof 10 liters per minute. The salt water in the tank is kept thoroughly mixed and is drained at arate of 5 liters per minute.

(a) Let y = y(t) be the amount of salt in the tank at time t. Give a differential equation relatingdydt

to y.

(b) Give a formula for the amount of salt in the tank at time t.

32. Find a set of parametric equations for the given equation:

x2

4+y2

49= 1

where the equations should be at (0,−7) when t = 0 and the curve should have a clockwiserotation.

33. (a) Finddy

dxand

d2y

dx2for the following set of parametric equations.

x = 4t3 − t2 + 7t y = t4 − 6

(b) Find the area between the x-axis and the curve traced out by

x(t) = 4t3 − t2, y(t) = t4 + 2t2, 1 ≤ t ≤ 3

34. (a) Convert the following equation from polar coordinates to rectangular coordinates.2

r= sin θ − sec θ

(b) Convert the following equation from rectangular coordinates to polar coordinates.4x

3x2 + 3y2= 6− xy

35. Set up but do not solve the integral that gives the area contained within the curve r = sin(3θ)?

0

Π

6

Π

3

Π

22 Π

3

5 Π

6

Π

7 Π

6

4 Π

3 3 Π

2

5 Π

3

11 Π

6

1.