Mat 116 – Business Calculus
Instructor: T. Bui
Name ________________________
Date ________________________
Test #1 (Chapter 10)
Please circle your final answer for each problem and show your work for partial credit.
1) Evaluate the following limits [15 points]
a) lim 5 x  4
x 1
b) lim
x 0
2x
x  2x 2
x2  9
x 3 x  3
c) lim
x2  x  2
d) lim
x 1
x 1
e) lim
x 5
x 5
x5
2) The following is a graph of f(x)
[14points]
A) lim  f ( x ) =
B) lim  f ( x) =
C) lim f ( x) =
D) f(-1) =
x  1
x  1
x  1
E) Is f continuous at 3?
F) Is f continuous at -2?
3) Use Graphing Calculator to find the following limits, given f ( x) 
A) lim f ( x ) =
x2
B) lim f ( x ) =
x2
C) lim f ( x ) =
4) List 3 things that a derivative represents [3points]
x 2
2x  2
( x  2) 2
[6points]
5) What is the limit definition of f’(x)
[2points]
6) Use the limit definition of f’(x) to find the derivative for f(x) = 3x + 4
[5points]
7) Use the limit definition of f’(x) to find the derivative for f(x) =
[5points]
x
8) Given f(x) = 3x2
A) Find the average rate of change of f(x) if x changes from 2 to 5
B) Find lim
h 0
f ( 2  h )  f ( 2)
h
C) What is the equation of the tangent line at x = 2
[3points]
[5points]
[3points]
9) The total sales of a company (in millions of dollars) t months from now are given by
S(t) = 2 t  10
[6points]
A) Find S(18) and interpret the result (you may use your calculator)
B) Find S’(18) and interpret the result (you may use your calculator)
10) The price-demand equation and the cost function for the production of table saws are giving,
respectively, by x = 6000 – 30p
and
C(x) = 72,000 + 60x
where x is the number of saws that can be sold at a price of $p per saw and C(x) is the total cost
(in dollars) of producing x saws.
[18points]
A) Express price p as a function of the demand x
B) Find the revenue function R(x) = xp using the result from part A)
D) Find the profit function in terms of x
E) Find the marginal profit (use shortcut)
F) How many table saws the company needs to produce and sell in order to have a maximum
profit (you may use your calculator or solve it algebraically)?
G) Find the average cost per unit if 500 saws are produced
11) Use the shortcuts to find the derivatives of the following functions [15points]
A) f(x) = 0.25x10
f’(x) =
B) f(x) = 3x4 – 8x3 + 5x + 2
f’(x) =
C) f(x) = 3x-1
f’(x) =
3x 5
2
D) f(x) =
f’(x) =
E) f(x) = 3x
5
3
f’(x) =
F) f(x) =
f’(x) =
3
x
Formula sheet for chapter 10:
Derivative shortcuts
• If f (x) = C, then f ’(x) = 0
• If f (x) = xn, then f ’(x) = n xn-1
• If f (x) = ku(x), then f ’(x) = ku’(x)
• If f (x) = u(x) ± v(x), then f ’(x) = u’(x) ± v’(x).
Profit function: P(x) = R(x) – C(x)
Revenue function: R(x) = xp
Marginal cost function: C’(x)
Marginal revenue function: R’(x)
Profit function: P’(x)
The exact cost of producing the (x + 1)st item is
C(x + 1) – C(x) or C’(x)
Average cost per unit: C ( x)  C ( x)
x
Average revenue per unit:
Average revenue per unit:
R ( x) 
R ( x)
x
P ( x) 
P ( x)
x