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SUNY College of Environmental Science and Forestry

APM 105: Survey of Calculus and Its Applications I

Written HW 2

Chapter 1

Complete the following problems in a neat and organized manner, with exercises in sequential order.

Make sure to label each problem (for example: 3(c)) and show all work where appropriate.

If you have more than one sheet, be sure to staple your papers.

(1) Given the two functions: f(x) = x2 g(x) = 12 + 2x � x2(a) Find the coordinates where f(x) and g(x) intersect.(b) Find the distance between the intersection points.

(c) Find the point halfway between the intersection points (the midpoint).

(d) Write an equation of the line that contains the points of intersection.

(2) It costs a certain manufacturer \$2,500 to make 700 units of a product, and \$4,000 to make 1200 units of the product.

The manufacturer will sell the product at a fixed price of \$50 per unit.

(a) What is the Cost function, assuming the relationship is linear (C = mx + b)?(b) What is the Revenue function?

(c) What is the Profit function?

(d) Find the number of units that need to be made and sold in order for the manufacturer to break even. Round

(3) When a pizza shop sells pizzas for \$15, they sell 700 pizzas per month. When the shop increased the price per pie

to \$18, they only sold 640 pizzas per month.

(a) What is the price (or demand) function, assuming the relationship is linear (p = mx + b)?(b) What is the Revenue function?

(c) How many pizzas that need to be sold to reach a revenue of \$5,000? Round to the nearest whole pizza.

(d) Find the cost function if the overhead cost of running the shop is \$4,000 per month, and the cost for ingredients

to make each pizza is \$2.50.

(e) Find the Profit function.

(f) Find the production level (number of pizzas made and sold) that would cause the shop to break even. Round

to the nearest whole pizza.

For the functions in (4) to (7):(a) Find the domain, in interval notation.(b) Find the y-intercept.(c) Find the x-intercept.

(d) Evaluate its di↵erence quotientf(x+h)�f(x)

h . Fully simplify the result to the point of canceling out the original h from the denominator.

(4) f(x) = 5x � 2 (5) f(x) = x2 � 4x (6) f(x) =p3x + 2

(7) f(x) =2

x

(8) Let f(x) =x2 + 4x + 4

x2 � 4.

(a) Find any discontinuities of f(x). For each, classify their type (removable or non-removable).(b) Find lim

x!�2f(x).

(c) Find limx!2

f(x).

(9) Let f(x) =2 � x

x2 � 8x + 12.

(a) Find any discontinuities of f(x). For each, classify their type (removable or non-removable).(b) Find lim

x!2f(x).

(c) Find limx!6

f(x).

I

.

(10) Sketch an example of a function f(x) with the following properties.(a) f(3) exists but lim

x!3f(x) does not exist.

(b) limx!3

f(x) exists but f(3) does not exist.

(11) Let g(x) =

px + 1 � 1

x.

(a) Find the domain of the function g(x) in interval notation.(b) Complete the table below, accurate to five decimal places. Use the result to estimate the limit in the center.

x �0.1 �0.01 �0.001 limx! 0

px + 1 � 1

x0.001 0.01 0.1

g(x)

(c) Find the limit limx!0

px + 1 � 1

xanalytically (algebraically, by hand). Show your work.

(12) Evaluate each of the limits algebraically, or show it does not exist.

(a) limx!2�

5 � xx + 2

(b) limx!16

px � 4

16 � x

(c) limx!0

14+x �

14

x

(d) limh!0

px + h �

px

h

(e) limx!�3+

2

x + 3

(f) limx!2�

x � 2|x � 2|

The graph of y = f(x) is shown below. Use it to answer questions (13) to (15).

5

-2

-1

1

2

3

4

y

x

-4 -3 -2 -1 1 2 3 4 5 6 7 8 9

(13) Visually determine the following limits and function values.

a) limx!�3+

f(x) b) limx!�3�

f(x) c) limx!�3

f(x) d) limx!�2+

f(x) e) limx!�2�

f(x) f) limx!�2

f(x)

g) limx!�1+

f(x) h) limx!�1�

f(x) i) limx!�1

f(x) j) limx!0+

f(x) k) limx!0�

f(x) l) limx!0

f(x)

m) limx!2+

f(x) n) limx!2�

f(x) o) limx!2

f(x) p) limx!4+

f(x) q) limx!4�

f(x) r) limx!4

f(x)

s) limx!5+

f(x) t) limx!5�

f(x) u) limx!5

f(x) v) f(0) w) f(2) x) f(4)

(14) Visually identify the x-values where f(x) is discontinuous. State the continuity condition each point violates.

(15) What new value should be assigned to f(2) to make the function continuous at x = 2?

(16) Let f(x) =

⇢x2 � 1, x  04x + 1, x > 0

. Sketch the graph of f(x) and discuss whether or not it is continuous.

For the functions in (17) to (19): Algebraically find any discontinuities, and state their type (removable or non-removable).

(17) f(x) =x + 2

x2 � 4 (18) f(x) =x4 � x3 � 6×2

x3 + 5×2 + 6x(19) f(x) =

4×2 � 16×2 � 11x + 4

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