Business Calculus I (Math 221) Exam 1

Business Calculus I (Math 221) Exam 1
October 3, 2012
Professor Ilya Kofman
Justify answers and show all work for full credit. No calculators allowed.
NAME:
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Problem 1 (14pts). The graph of y = f(x) is shown above. Evaluate each limit, or write
DNE if the limit does not exist. No justifications are necessary.
(a) lim
x→5 + f(x) =
(b) lim
x→5 − f(x) =
(c) lim
x→4
f(x) =
(d) lim
x→3 − f(x) =
(e) lim
x→3 + f(x) =
(f) lim
x→1
f(x) =
(g) For f(x) to be continuous at x = 4, we must set f(4) =

Problem 2 (12pts). Evaluate these limits. If a limit does not exist (DNE), you must justify.
Show all work!
(a) lim
x→4
x 2 −x−12
x−4
(b) lim
x→2
x−2
|x−2|
(c) lim
x→∞
5x 3 −3x 2 +10
−4x 3 +x 2
Problem 3 (6pts). For what value of c (if any) is the function f(x) continuous at x = 1?
Justify your answer. ⎧
⎪⎨ x+ 3 x < 1
x−2
f(x) = c x = 1
⎪⎩
x 2 −3x x > 1

Problem 4 (8pts). Let f(x) = x 2 −5x. Use the definition of the derivative to find f ′ (1).
Problem 5 (30pts). Compute the derivative dy . Do not simplify. Show all work!
dx
(a) y = 5 3√ x− 1 x 2
(b) y = 5√ x 3 +8x−3
(c) y = x4 +3x
x 2 −1
(d) y = (2x 5 −7x)(3x 2 +6x+2)
(e) y =
(
9+ √ ) 14
x 2 +4

Problem 6 (12pts). Let g(x) = 1
2x+3 .
(a) Find g ′ (1).
(b) Find g ′′ (1).
Problem 7 (8pts). Let F(x) = x 2 + 7x − 3. Find the equation of the tangent line to the
graph of F(x) at x = −1. Leave your answer in the form y = mx+b.
Problem 8 (6pts). Given the graph y = f(x) below, sketch the graph y = f ′ (x) below.

Problem 9 (15pts). For x units sold, the total revenue function is R(x) = 40x+2000. The
total cost function is C(x) = 1000+25x+ 1 10 x2 .
(a) Find the profit function P(x).
(b) Find the marginal profit when 100 units are sold.
(c) Should the company sell more than or fewer than 100 units? Explain.