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Math 135 Exam 1 Study Guide 1

Sect. 1.1

1: Consider the function H given by

H(x) =

  2x + 2 for x < 1

2x− 4 for x ≥ 1

a) Graph the function.

b) Find lim x→1

H(x) both numerically and graphically.

c) Find lim x→−3

H(x) both numerically and graphically.

2: Calculate the following limits based on the graph of f.

a) lim x→2−

f(x)

b) lim x→2+

f(x)

c) lim x→2

f(x)

Sect. 1.2

3: Find lim x→0

√ (x2 − 3x + 2).

4: Find the following limits and note the Limit Property you use at each step:

a) lim x→1

2x3 + 3x2 − 6

b) lim x→4

2x2 + 5x− 1 3x− 2

c) lim x→2

√ 1 + 3x2

Math 135 Exam 1 Study Guide 2

5: Find lim x→−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

g(x) =

 

1

2 x + 3 for x < −2

x− 1 for x ≥−2

continuous at x = −2? Why or why not?

8: Is the function g given by

h(x) =

 

x2 − 9 x− 3

for x 6= 3

7 for x = 3

continuous at x = 3? Why or why not?

9: Let

p(x) =

 

x2 − 25 x− 5

for x 6= 5

c for x = 5

Determine c such that p is continuous at x = 5.

Sect. 1.3

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.

Sect. 1.4

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).

Math 135 Exam 1 Study Guide 3

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.

Sect. 1.5

14: Find each of the following derivatives:

a) d

dx (5x3 − 7)

b) d

dx

( 24x−

√ x +

5

x

)

c) d

dx

( 3x5 + 2 3

√ x +

1

3x2 + √

5

) 15: Find the points on the graph of f(x) = −x3 + 6x2 at which the tangent line is horizontal.

Sect. 1.6

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.

Math 135 Exam 1 Study Guide 4

Sect. 1.7

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.

Sect. 1.8

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.