Practice Problems
1. If the total profit, in thousands of dollars, for a product is given by
√
P (x) = 10 x + 1 − 3x − 14,
what is the marginal profit at a production level of 8 units?
2. The revenue from the sale of x tables is represented by the following
formula.
R = 20(3x + 1)−1 + 60x − 15 dollars
Find the marginal revenue when 3 tables are sold.
3. Given y = x3 −6x2 +3x−5. Find the critical values of the function. Make
a sign diagram and determine the relative maxima and minima. Determine
absolute maxima and minima on interval [2, 4].
4. Is the graph of f (x) = x3 − 4x2 + 3 concave up or down at the point
(1, 0)?
5. Use properties of limits and algebraic methods to find the limit, if it
exists.
a) lim 3x25x+2
+2x−4
x→2
2
2
b) lim 4(x+h)h −4x
x→0
6. Find the derivative of the function.
a) y = ln(x10 )
b) y = (ln x)7
7. Find the derivative of the function.
a) y = ln(5x3 + 3x) + 5x2 + 18
3
2
4
b) y = 5e(x +5x −7x) + 10x2 − 3
8. Find the derivative of the function
a) y = e2x (x − 2)3 − 57x + 3
2
−1
b) y = x3x+4x−5
9. Find the first four derivatives of f (x) = x5 − 2x4 + 3x2 − 11x + 2 − e5x
10. Find dy/dx for the function.
x2 + 12y 2 − 9 = 0
11. Let y = x5 − 6x + 1. Find dy/dt given that x = 2 and dx/dt = 3.
12. Evaluate the integral. Check your answer by differentiating. (Use C for
the constant of integration.)
Z 1
3
x7 − 6 + √
+ e5x dx
4
x
x−1
1
13. Use algebra to rewrite the integrand; then integrate and simplify. (Use
C for the constant of integration.)
Z
(3x2 + 2)2 x4 dx
14. Evaluate the integral. Check your result by differentiation. (Use C for
the constant of integration.)
Z
3x4 dx
(7x5 − 4)4
15. Cost is in dollars and x is the number of units. If the marginal cost for
a product is M C = 6x + 3, and the production of 10 units results in a total
cost of $520, find the total cost function.
16. In this problem, use properties of definite integrals.
R0
R 11 3
R 11 3
14641
x
If −11 x3 dx = −14641
and
dx
=
,
what
does
0
−11 x dx equal?
4
4
17. Suppose that the cost in dollars for x chairs is given by
C(x) = 400+x+0.3x2 . What is the average value of C(x) for 10 to 20 units?
18. Equations are given whose graphs enclose a region. Find the area of
the region.
f (x) = x3 − x2 , g(x) = 2x.
R∞ dt
19. Evaluate the improper integral t3/2
20. Evaluate the improper integral
1
−1
R
−∞
4
x dx
21. In this problem, p is in dollars and x is the number of units.
Find the producer’s surplus for a product if its demand function is
p = 169 − x2 and its supply function is p = x2 + 8x + 127.
22. Evaluate the function as indicated.
z(x, y) = (5x − 1)ex−y ; find z(0, −1)
23. Give
the
√ domain of the function.
3x5y − y+11x
z=
x+2y
∂z
∂z
24. If z = x3 + 5x2 − 3x + 7y 3 − 2y − 11, find ∂x
and ∂y
.
2
y
2
25. If z = xy − 3xe + y , find the following.
a) zxx =
b) zxy =
c) zyx =
d) zyy =
2