1. Find the following limits. Include“∞” as appropriate.
Show work as necessary.
x2 − x
a. lim
x→5 x + 5
x2 + 3x
b. lim
x→−3 x + 3
x − x2
c. lim
x→4 x − 4
2. The graph of y = f (x) is given below. Find the indicated limits if they
exist. If a limit doesn’t exist, say so.
a. lim f (x)
x→4
c.
lim f (x)
x→−2
b. lim f (x)
x→2−
d. lim f (x)
x→2+
2
3. Write an equation for the tangent line to the curve y = 2x −
3
when x = 3.
x
2x3
.
4. Find vertical and horizontal asymptotes of the graph of y =
x(1 + 2x)2
5. Find the second derivative of f (x) =
4
. Simplify appropriately.
3x − 1
6. Find the first-order partial derivatives of f (x, y) = xey + y 2 ln x − 2.
3
7. The derivative of the function f (x) is f ′ (x) =
x(x + 2)
.
2x − 4
a. State the intervals over which f is increasing.
b. State the intervals over which f is decreasing.
c. Find the x-coordinate of any relative extrema.
8. The function f , defined for all real values, satisfies the following properties:
(a) f (−4) = 0, f (−2) = 2, f (0) = 4, f (2) = 1, f (3) = −2, f (4) = 0
(b) lim f (x) = ∞, lim f (x) = 2
x→−∞
< 0,
> 0,
< 0,
(d) f ′′ (x)
> 0,
(c) f ′ (x)
x→∞
on (−∞, −4) and (0, 3)
on (−4, 0) and (3, ∞)
on (−2, 2) and (4, ∞)
on (−∞, −2) and (2, 4)
1. Compute these integrals
Z
a.
6x e−3x dx
b.
Z
e
x2 ln x dx
1
12
on the interval [1, 2].
2. Find and identify absolute extrema of f (x) = 3x2 +
x
Give both coordinates.
2
3. Let f (x, y) = 8x3 − 6xy + y 2 . Find the critical points of f and identify each
as one of: relative min, relative max, or saddle point.
4. Use Lagrange multipliers to find the maximum of f (x, y) = xy + 3y subject
to the constraint y + 2x = 10.
1. John is selling wine at $30 per bottle. However, for orders beyond 100
bottles the price of every bottle is reduced by 5c
/ for each extra bottle
beyond 100.
Write a formula for the revenue function R(x) and use calculus methods to
find the number of bottles to sell that will maximize revenue. What would
be the maximum revenue possible?
2. Jeana is building a box for her pets. It will have a square top, but no
bottom and only three sides. If the material costs $6 per square foot for the
top and $2 per square foot for the sides, what’s the largest box (maximum
volume) that can be built on a budget of $288 ?
2
3. In a certain economics model, the revenue function is R(x) = 2x(x2 − 3x)
2
3
and the cost function is C(x) = x3 − x2 + 10.
3
2
Find the marginal revenue, marginal cost and marginal profit functions.
4. In a certain economics model, the marginal revenue function is 10x +
and the marginal cost function 2(5 − e−x/2 ).
a. Find the total cost for 20 items (i.e., from 0 to 20).
b. Find the revenue generated by the items sold from 2 to 10.
40
x2