Math 142-202
Sample Questions for Final Exam
Instructor: Minh Kha
1. Find the domain of the function f (x) = √
ex
+ ln (x − 1).
x2 − 1
2. The producer is not willing to sell any item if the price is set at $10. However, the producer will supply 5 items
when the price is set at $20. Find the supply equation.
3. Solve the equation
• 5x = 25x+2
• 2xex = x2 ex + ex
4. What amount will an account have after 10 years if $1000 is invested at an annual rate of 10% compounded
continuously?
5. Given logx y = 5 and logx z = 4, evaluate logx (xy 2 z 3 ).
6. Solve the equation ln (4x) = 2 ln (8x)
7. Find the value of A so that the limit will have the indicated value.
lim Ax2 + 2Ax + ex−1 = 1
x→1
8. Find the value of the constant c that makes h continuous on (−∞, ∞).
h(x) =
2ex + c2
2c(x + 1)2015 + e10x
x<0
x≥0
9. Compute the average rate of change of the function f (x) = x2 over the interval [−1, 1].
2
10. Compute the instantaneous rate of change of the function f (x) = x2 at the point x = −1.
11. What is the equation of the line tangent to f (x) = x2 at x = 1?
12. Evaluate the following limits
x2 − 9
.
x→3 2x − 6
• lim
2
e−x
• lim
x→∞ x + 1
13. How can a function fail to be differentiable? List possible reasons with examples.
3
14. Given g(x) = ln (f (x)), f (2) = 10, f 0 (2) = 1. What is g 0 (2)?
15. Compute the derivatives of the following functions:
• e2x
2
+x+1
• ln(x2 + 1)
16. A company sells hats at $15 each and calculates the demand function for the hats to be D(p) = 100 − 5p
• Is the demand elastic or inelastic at the current $15 price?
• Should the hat price be raised or lowered to increase revenue?
• Set E(p) = 1 and solve for p to determine at which price the revenue is greatest.
4
17. Consider the function f (x) = x4 − x3 . Find all the inflection points of f .
18. Suppose the function f is defined on the whole real line and its derivative is shown as follows:
f 0 (x) = ex (2x − 1)7 .
Find the intervals where f is increasing.
19. Find the absolute minimum and maximum of the function f (x) = −x2 − 2x + 10 on the interval [0, 1].
20. The price and cost in dollars for x items are given by p(x) = 200 − 2x and C(x) = −x2 + 100x + 150. Determine
the production level x that will leave the maximum profit.
5
21. What is the average value of the function f (x) = x over the interval [−1, 1]?
22. Find the right Riemann sum R2 of the function f (x) = x defined on the interval [0, 2].
23. Assume that f (x, y) =
√
1
√
+ x + 2 y. Find the domain of f .
xy
24. Let f (x, y) = −x2 + 2xy − y 2 + 1.
(a) Compute the first derivatives fx (x, y) and fy (x, y).
(b) Find all the second derivatives fxx (x, y), fyy (x, y), and fxy (x, y).
(c) Use the second derivative test for function of two variables to locate and classify all the critical points of
f (x, y).
6
Use the below graph of the derivative f 0 (x) to obtain answers.
Z
25. Find
5
f 0 (x)dx.
0
26. Suppose that f (0) = 1. Compute f (5).
27. Find all critical points of f , and find local extremum of f
7
28. Compute the following integrals:
Z b
2
•
(2x + 1)ex +x dx
a
Z
x3
dx
x4 + 1
Z
(ln x)8
dx
x
•
•
8
29. Sketch the graph of a continuous function f such that the following conditions hold:
• Domain: (−∞, 2) ∪ (2, ∞)
• Vertical asymptotes when y → −∞: x = 2.
• No horizontal asymptotes.
• f 0 (x) < 0 on (−∞, 2) and f 0 (x) > 0 on (2, ∞).
• f (0) = f (4) = 0.
• f 00 (x) < 0 on (−∞, 2) ∪ (2, ∞).
9