1: Consider the function H given by
2x + 2 for x < 1
H(x) =
2x - 4 for x ≥ 1
a) Graph the function.
b) Find limx→1 H(x) both numerically and graphically.
c) Find limx→-3 H(x) both numerically and graphically.
2: Calculate the following limits based on the graph of f .
a) limx→2- f(x)
b) limx→2+ f(x)
c) limx→2 f(x)
3: Find limx→0 √ (x2 - 3x + 2).
4: Find the following limits and note the Limit Property you use at each step:
a) limx→1 2x3 + 3x2 - 6
b) limx→4 2x2 + 5x - 1/3x - 2
c) limx→2√(1 + 3x2)
5: Find limx→-3 (x2 - 9/x + 3).
6: Is the function f given by f (x) = x2 - 5, continuous at x = 3? Why or why not?
7: Is the function g given by
1/2x + 3 for x < 2
g(x) =
x - 1 for x ≥ -2
continuous at x = -2? Why or why not?
8: Is the function g given by
(x2 - 9/x - 3) for x ≠ 3
h(x) =
7 for x = 3
continuous at x = 3? Why or why not?
9: Let
(x2 - 25/x-5) for x ≠ 5
p(x) =
c for x = 5
Determine c such that p is continuous at x = 5.
10: State the average rate of change for each situation in a short sentence. Be sure to include units.
a) It rained 4 inches over a period of 8 hours.
b) Your car travels 250 miles on 20 gallons of gas.
c) At 2 P.M., the temperature was 82 degrees. At 5 P.M., the temperature was 76 degrees.
11: For f (x) = x2, find f'(x). Then find f'(-3) and f'(4).
12: For f (x) = x3, find f'(x). Then find f'(-1) and f'(1.5).
13: For f (x) = 3/x:
a) Find f'(x).
b) Find f'(2).
c) Find an equation of the tangent line to the curve at x = 2.
14: Find each of the following derivatives:
a) d/dx(5x3 - 7)
b) d/dx(24x - √x +(5/x))
c) d/dx(3x5 + 23√x +(1/3x2) + √5)
15: Find the points on the graph of f (x) = -x3 + 6x2 at which the tangent line is horizontal.
16: Find:
a) d/dx[(x4 - 2x3 - 7) (3x2 - 5x)]
b) d/dx [(2x5 + x - 1) (3x - 2)]
c) d/dx [(√x+1)(5√x - x)]
17: Differentiate and simplify your result:
a) f (x) = x2 - 3x/ x - 1
b) g(x) = 1 - 3x/x2 + 2
18: Paulsen's Greenhouse finds that the cost, in dollars, of growing x hundred geraniums is given by C(x) = 200 + 100 • 4√x. If the revenue from the sale of x hundred geraniums is given by R(x) = 120 + 90 • √x, find each of the following.
a) The average cost, the average revenue, and the average profit when x hundred geraniums are grown and sold.
b) The rate at which average profit is changing when 300 geraniums are being grown.
19: Differentiate f (x) = (1 + x3)1/2.
20: Differentiate f (x) = (3x - 5)4(7 - x)10.
21: Differentiate f (x) = 2x2 - 1/(3x4 + 2)2.
22: GameBoss Video's profit, in dollars, is given by y = f (u) = 0.1u2 - 500, where u is the number of units sold. Its sales are given by u = g(x) = 125x + 40, where x is the number of days. Find the rate at which profit is changing on the 5th day.
23: For y = 1/x, find d2y/dx2.
24: For y = (x2 + 10x)20, find and simplify y' and y''.
25: Find y'':
a) y = -6x4 + 3x2
b) y = 2/x3
c) y = (3x2 + 1)2
26: For s(t) = 10t2 find v(t) and a(t), where s is the distance from the starting point, in miles, and t is in hours. Then, find the distance, velocity, and acceleration when t = 4 hr.