Business Calculus : Rate of Change, Slope of Tangent and Limit Concept (12 Problems)

1. Consider the following table:

X 0 1 2 3 4 5
Y 1.5 2.4 3.6 4.5 5.6 6.7

For each of the following intervals, find the average rate of change:

(a) [0,1]
(b) [0,3]
(c) [2,5]
(d) [0,5]

2. Consider the following table:

X 1 2 3 4 5 6
Y 24 26 28 27 25 23

For each of the following intervals, find the average rate of change:

(a) [1,3]
(b) [2,4]
(c) [2,6]
(d) [1,5]

14. A model for the percentage change in the Consumer Price Index for medical care is as follows:

Verbal Definition: I(x) = the percentage change in the Consumer Price Index for medical care x years after Dec. 31st, 1986 (for the preceding year).
4 3 2
Symbol Definition: I(x) = 0.0193x - 0.3547x + 1.9314x - 3.0161x + 7.6229, x is greater than or equal to 0.

Assumptions: Certainty and divisibility. Certainty implies that the relationship holds exactly. Divisibility implies fractions are possible for both variables.

(a) For each of the following intervals, calculate the average rate of change: [0,1] [2,4].
Express your answers in complete sentences.
(b) What is the percent rate of change over these intervals? Express your answers in complete sentences.

2. (a) Using technology and the limit concept, estimate the slope of the tangent line to the curve :
3
y = f(x) = x - 3x + 2 at the point (1,0).

(c) Sketch a graph of the curve and the tangent line.

3. Using technology and the limit concept, estimate the slope of the tangent line to the curve:
2
y = x + 2 at x = 3.

2
1. If y = f(x) = 5x - 2x + 2,0 is less than or equal to x, which is less than or equal to 3, and f'(x) = 10x - 2, 0 is less than or equal to x, which is less than or equal to 3,

(a) Find f(1), f (2.5), and f(3).
(b) Find f' (1). Use this to estimate the change in y if x increases by 0.1 to 1.1, and then use this answer to approximate f (1.1). Now calculate the actual exact value of f (1.1). How far off is your approximation?

(c) Find df
---- when x = 2.5.
dx

(d) Find dy
---- when x = 3.
dx
2
2. If y = g(x) = -2x + 4x - 10, - 10 is less than or equal to x, which is less than or equal to 10, and g'(x) = -4x + 4, -10 is less than or equal to x, which is less than 10,

(a) Find g'(3), g' (-10), and g'(0).
(b) Find g (3), g (-10), and g(0).

(c) Find dg
---- when x = -2.
dx
(d) Find dy
---- when x = 4.
dx
(e) Use your answers from part (a) and part (b) for g(3) and g'(3) to estimate the change in the value of g(x) if the value of x increases from 3 to 3.1, and use this answer to approximate the value of g (3.1). Now calculate the actual exact value of g (3.1). How far off is your approximation?

5. Rabbits are very prolific. You have been raising rabbits and determined that after buying one pair for breeding, you can model the number of rabbits that you have by:

Verbal Definitions: R(t) = the number of rabbits t years after the start of breeding.
2
Symbol Definition: R(t) = 5t - 2t + 2,0 is less than or equal to t, which is less than or equal to 3.

Assumptions: Certainty and divisibility. Certainty implies the relationship is exact. Divisibility implies that any fractions of rabbits and years are possible.

(a) How many rabbits will you have after one year? Write your answer in a complete sentence.
(b) How many rabbits will you have after two and a half years? Write your answer in a complete sentence.
(c) How many rabbits will you have after three years? Write your answer in a complete sentence.
(d) Use technology and the limit concept to estimate how fast the number of rabbits is changing after the one year. Write your answer in a complete sentence.
(e) Use technology and the limit concept to estimate how fast the number of rabbits is changing after two and a half years. Write your answer in a complete sentence.
(f) Use technology and the limit concept to estimate how fast the number of rabbits is changing after three years. Write your answer in a complete sentence. What is happening to the rate of change over time?
(g) Use your answers to parts (c) and (f) to approximate the number of additional rabbits you would expect between t = 3 and t = 3.5, and then use this answer to approximate the total number of rabbits 3.5 years after breeding. How far off is your approximation?

6. Recall the ball toss problem from the beginning of this section. The model was:

Verbal Definition: H(t) = the height in feet of the ball t seconds after it was tossed.
2
Symbol Definition: H(t) = -16t + 48t + 5, 0 is less than or equal to t, which is less than or equal to 3.

Assumptions: Certainty and divisibility. Certainty implies the relationship is exact. Divisibility implies that any fractions of time and feet in height are possible.

(a) Use technology and the limit concept to estimate H'(0.5). Write your answer in a complete sentence.

(b) Use technology and the limit concept to estimate dH
---- when t = 1.5.
dt
Write your answer in a complete sentence.

(c) If we can define y = H(t), use technology and the limit concept to estimate dy
---
dt
when t = 2. Write your answer in a complete sentence.

Exercises for Section 2.4, p. 242-243, Questions: 2,3,10, and 11
? Exercises for Section 2.5, p. 261-262, Questions: 2,3,15, and 16
? Exercises for Section 2.6, p. 274-275, Questions: 1,2,11, and 12

2. Use the definition of the derivative to find the derivatives of the following functions:
(a) y = -3
(b) V(t) = 0.45
(c) y = x -7
(d) C(x) = -0.01x + 12
2
(e) y = 4.5x + 5
2
(f) h(t) = -3t + 8t + 7

3. Use the definition of the derivative to find the derivatives of the following functions:
(a) y = 5
(b) t = 3.08
(c) y = f(x) = 3x
(d) r(q) = 0.8q

10. The model for the profit from the sale of "Save the Environment" mugs is:

Verbal Definition: pie(q) = profit in dollars from the sale of q mugs.
2
Symbol Definition: pie(q) = -0.01q + 6.5q - 25, 0 is less than or equal to q, which is less than or equal to 850.

Assumptions: Certainty and divisibility. Certainty implies that the relationship is exact. Divisibility implies that any fractions of mugs and dollars of profit are possible.

Find the value of the marginal profit (rate of change of profit with respect to quantity) function when:

(a) 200 mugs are sold.
(b) 300 mugs are sold.
(c) 400 mugs are sold.
(d) What conclusions can be drawn about your goals for sales?

11. You have been selling sodas at sporting events and have developed the following model for the demand:

Verbal definition: q(p) = the quantity of sodas (in thousands) sold at p dollars per soda.
2
Symbol definition: q(p) = --- , 0.5 is less than or equal to p, which is less than or
p
equal to 5.

Assumptions: Certainty and divisibility. Certainty implies that the relationship is exact. Divisibility implies that any fractions of sodas and dollars of selling price are possible.

Find the rate at which the quantity of sodas sold was changing when the price was:

(a) $1 per soda.
(b) $2 per soda.
(c) $3 per soda.
(d) Describe how the demand is changing as the price increases.
2 dy
2. If y = (3 - 5x), then ----- = ______.
dx

3. d 2
--- square root of 25 + x = ______.
dx

15. For the demand data for selling sweatshirts given in Sample Problem 8 of Section 1.3:

Selling Price $10 $15 $20 $25
Quantity Sold 40 25 13 5

(a) Fit an exponential function to obtain a model of the quantity sold as a function of the selling price, q = D(p), and fully define this model.
(b) Find the derivative of your function at $20 and interpret your answer in words.
(c) Calculate D(20) and use this value and your answer to part (b) to estimate the demand at $22.

16. In Sample Problem 4 of Section 1.5, we found that we could model the percentage of U.S. households with cable TV as follows (rounded to 3 significant digits):

Verbal Definition: P(x) = the percentage of U.S. households with cable TV x years after Dec. 31st, 1977.

65.7
Symbol Definition: P(x) = ------------- , 0 is less than or equal to x, which is less than or equal to 21.
-0.262x
1 + 3.53 e

Assumptions: Certainty and divisibility. Certainty implies that the relationship is exact. Divisibility implies that any fractions of years or percentage points are possible.

(a) Find the derivative of this function at x = 10, and interpret your answer in words.
(b) Calculate P(10), and use this value and your answer to part (a) to estimate the percentage of U.S. households with cable TV at the end of 1989.

For exercises 1-2 below, find the derivatives of the following functions, using the product rule. Where possible, check your answers by multiplying the functions and using one of the polynomial rules.

1. y = f(x) = x (3x + 5)
2. g(r) = r(-2r + 5)

11. You have been selling sodas at local sporting events. The number of sodas that you sell depends on several different things, such as the weather (you sell more on hot dry days) and the attendance. These are things over which you have no control, but you can change the price that you charge. You have collected data over a period of time and tried to keep the other variables such as weather and popularity of team as constant as possible. You have found that you can model the number of sodas sold as a function of the price charged per soda:

Verbal Definition: N(p) = the number of sodas (thousands) sold in one day at p dollars per soda.
p
Symbol Definition: N(p) = 1.0924(0.3178) , 0.5 is less than or equal to p, which is less than or equal to 5.

Assumptions: Certainty and divisibility. Certainty implies that the relationship is exact. Divisibility implies that any fractions of numbers of soda and dollars are possible.

Find the rate at which the number of sodas sold is changing when the price is:

(a) $1.00 per can.
(b) $3.00 per can.
(c) $5.00 per can.
(d) What conclusions, if any, could you draw from these answers?

12. Using the model in Exercise 11, find a model for the revenue from the sale of sodas as a function of the price charged. Find the rate of change of the revenue with respect to the price charged and interpret your answer when soda is sold at:
(e) $1.00.
(f) $3.00.
(g) $5.00.
(h) What conclusions, if any, could you draw from these answers?