2.5 Solving Problems Involving Rates of Change
Chapter
Chapter 2
Section
2.5
Lectures 6 Videos
Solutions 23 Videos

The cost of running an assembly line can be modeled by the function C(x) = 0.3x^2 - 0.9x - 1.675, where C(x) is the cost per hour in thousands of dollars and x is the number of item produced per hour in thousands. The most economical production level occurs when 1500 items are produced. What is the rate of change when x = 1500?

3.06mins
Q1

For a person at rest, the function P(t) = -20 \cos(300^{o}t)+ 100 models blood pressure, in millimetres of mercury (mm Hg), at time t seconds. What is the rate of change in blood pressure at 3 s?

1.00mins
Q2

If a function has a maximum value at (a, f(a)) , what do you know about the slopes of the tangent lines at the following points?

(a) point to the left of, and very close to, (a, f(a))

(b) point to the right of, and very close to, (a, f(a))

0.41mins
Q3

If a function has a minimum value at (a, f(a)) , what do you know about the slopes of the tangent lines at the following points?

(a) point to the left of, and very close to, (a, f(a))

(b) point to the right of, and very close to, (a, f(a))

0.31mins
Q4

For each function, the point given is the maximum of minimum. Use the difference quotient to verify that the slope of the tangent at this point is zero.

f(x) = 0.5x^2 + 6x + 7.5; (-6, -10.5)

2.38mins
Q5a

For each function, the point given is the maximum of minimum. Use the difference quotient to verify that the slope of the tangent at this point is zero.

f(x) = -6x^2 + 6x + 9; (0.5, 10.5)

1.22mins
Q5b

For each function, the point given is the maximum or minimum. Use the difference quotient to verify that the slope of the tangent at this point is zero.

\displaystyle f(x) = -4.5\cos(2x);  \displaystyle (0^o, -4.5) 

Q5d

Use an algebraic strategy to verify that the point given for the function is either a maximum or minimum.

f(x) = x^2 - 4x + 5 at (2, 1)

Q6a

A pilot who is flying at an altitude of 10 000 ft is forced to eject from his airplane. The path his ejection seat takes is modelled by the equation h(t) = -16t^2 + 90t + 10000, where h(t) is his altitude in feet and t is the time since his ejection in seconds. estimate at what time, t, the pilot is at a maximum altitude. Explain how the maximum altitude is related to the slope of the target line at certain points.

Q7

The top of a flagpole sways back and forth in high winds. The function f(t) = 8\sin(180^ot) represents the displacement, in centimetres, that the flagpole aways from vertical, where t is the time in seconds.

The flagpole is vertical when f(t) =0. It is 8 cm to the right of its resting place when f(t) = 8, and 8 cm to the left of its resting place when f(t) = -8. If the flagpole is observed for 2s, it appears to be furthest to the left when t= 1.5 s. Is this observation correct?

Justify your answer using the appropriate calculations for the rate of change in displacement.

Explain how to determine the value of x that gives a maximum for a transformed sine function in the form y = a\sin(k(x-d)) + c, if the maximum for y = \sin x occurs at (90^o, 1)