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Mat 116 – Business Calculus
Prof. Bui
Name ________________________
Date ________________________
Midterm (chapter 10 and 11)
Please circle your final answer for each problem and show your work for partial credit.
1) Given f(x) as a function below.
a) Find lim f ( x)
b) Find lim f ( x ) ?
b) Find lim f ( x)
d) Find f(2) =
x2
x 2
x2
2) Determine the limits, using graph or table
a) lim 
x  2
1
x2
3) f(x) =
b) lim 
x  2
1
x2
1
x  2 x  2
c) lim
2x
3x  1
Is f(x) continuous?
If not, find the point of discontinuity ________
4) f(x) = x2 -2x
Is f(x) continuous?
If not, find the point of discontinuity ________
5) Find the limits
a) lim 3x  5
2
x
 4x4  x2  6
b) lim
x 
x5
6) Evaluate the following limits
Note: when you find the limits of these problems, you must factor first if possible and then
simplify before you substitute the number for x
a) lim
x3
4x  3
2
x  1 x  3 x 2
b) lim
x 2  49
c) lim
x 7 x  7
x 2  2x  3
d) lim
x3
x 3
7) Given f(x) = 3x2
a) Use the shortcut to find the derivative of f(x)
f’(x) =
b) What is the equation of the tangent line at x = 1
c) Find the value(s) of x where the tangent line is horizontal
8) The price-demand equation and the cost function for the production of graphing calculators
are giving, respectively, by x = 5000 – 25p
and
C(x) = 70,000 + 50x
where x is the number of calculator that can be sold at a price of p per calculator and C(x) is the
total cost (in dollars) of producing x calculators
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)
C) Find the profit function in terms of x (formula: P(x) = R(x) – C(x))
9) Use the shortcuts to find the derivatives of the following functions. Circle your answers.
A) f(x) = 3.2x10
f’(x) =
f’(x) =
C) f(x) = 8x
3x 8
D) f(x) =
2
-2
f’(x) =
E) f(x) =
f’(x) =
 7x
2
7
f’(x) =
G) f(x) = 
f’(x) =
B) f(x) = 5x4 – 7x3 + 3x2 – x + 11
F) f(x) =
4
x
f’(x) =
1 x
e
6
H) f(x) = 4 ln x
f’(x) =
I) f(x) = 2 x  e x  ln x
J) f(x) = ln x3
f’(x) =
f’(x) =
K) f(x) = log 4 x
L) f(x) = 9 x
f’(x) =
f’(x) =
M) f(x) = x + ln (5x)
N) f(x) = x4 ex [factor your final answer]
f’(x) =
f’(x) =
O) f(x) =
f’(x) =
3x  2
[simplify]
x3
P) f(x) = ( x12  9) 2
f’(x) =
[simplify]
FORMULA SHEET
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).
• If f (x) = ex, then f ’(x) = ex
1
• If f (x) = ln x, then f ’(x) =
x
• If f (x) = ax, then f ’(x) = ax ln a
• If f (x) = log a x , then f ’(x) =
1
x ln a
• If f (x) = U · V, then f ’(x) = U’ V + V’ U
• If f (x) =
U
, then f ’(x) =
V
U ' V  V 'U
V2
• If f (x) = kUn, then f ’(x) = kn Un-1 U’
Profit function: P(x) = R(x) – C(x)
Revenue function: R(x) = xp
Marginal cost function: C’(x)
Marginal revenue function: R’(x)
Marginal Profit function: P’(x)
Product Rule
Quotient Rule
General Power Rule
Chain Rule