Fall 2015
Math 1325 – Test #1 Review – Chapter 3
1. Determine if each limit exists. If the limit exists, find its value.
Graph A
Graph B
a) lim f (x)
a) lim f (x)
b) lim f (x)
b) lim f (x)
x 0
x 0
x 2
x 
2. Suppose lim f ( x)  9 and lim g(x)  27 . Use the limit rules to find each limit.
x 4
x 4
a) lim  f (x)  g(x)
x 4
b) lim
x 4
f ( x)
g(x)
c) lim
x 4
f ( x)
3. Use the properties of limits to decide whether each limit exists. If the limit exists, find its value.
x2  9
x 3 x  3
x 5
c) lim
x 25 x  25
3x 2  2 x
e) lim
x   2 x 2  2 x  1
a) lim
 x  1, if x  3

4. Let f (x)  2,
if 3  x  5 .
 x  3, if x  5

5x 2  7 x  2
x 1
x2  1
3x
d) lim
x  7x  1
2x 3  x  3
f) lim
x  6x2  x  1
b) lim
Find lim f ( x) and lim f ( x) .
x 3
5. Determine if the following function is continuous at x = 0.
x 5
if x  0
4 x  4,
f ( x)   2
 x  4 x  4, if x  0
Fall 2015
6. Find all values of x = a where the function is discontinuous. For each point of discontinuity, give the
following:
f(a) ; lim f (x) ; lim f (x) ; lim f (x) ; Explain why the function is discontinuous at a.
x a
x a
x a
7. Find the average rate of change of the function y = x2 + 2x over the interval [1, 3].
8. Find the average rate of change of the function y = –3x3 + 2x2 – 4x + 1 over the interval [–2, 1].
9. Find the instantaneous rate of change of the function f(x) = x2 + 2x at x = 0.
10. Find the instantaneous rate of change of the function f(x) = 1 – x2 at x = –1.
11. Suppose that the total profit (in hundreds of dollars) from selling x items is given by:
P(x) = 2x2 – 5x + 6
a) Find the marginal profit when x = 2. Interpret using a complete sentence.
b) Find the marginal profit when x = 4. Interpret using a complete sentence.
12. Suppose customers in a hardware store are willing to buy N(p) boxes of nails at p dollars per box,
as given by:
N(p) = 80 – 5p2, 1 ≤ p ≤ 4.
a) Find the average rate of change of demand for a change in price from $2 to $3.
b) Find and interpret the instantaneous rate of change of demand when the price is $2.
13. Using the definition of the derivative, find f ’(x).
a) f ( x ) 
12
x
b) f (x)  x
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14. For each function, find the equation of the tangent line at the given value of x.
a) f(x) = x2 + 2x ; x = 3
b) f(x) = 6 – x2 ; x = –1
c) f(x) = 4 x ; x = 16
15. Find the values of x where the function does NOT have a derivative.
16. Suppose the demand for a certain item is given by D(p) = –2p2 – 4p + 300 where p represents the
price of the item in dollars.
a) Find the rate of change of demand with respect to price.
b) Find and interpret the rate of change of demand when the price is $10.
17. Waverly Products has found that its revenue is related to advertising expenditures by the following
function where R(x) is the revenue (in dollars) when x hundred dollars are spent on advertising.
R(x) = 5000 + 16x – 3x2
a) Find the marginal revenue function
b) Find and interpret the marginal revenue when $500 is spent on advertising.
18. For each of the following, determine which graph is the function and which graph is the derivative.
a)
b)