Chapter Review on Rates of Change
Chapter
Chapter 2
Section
Chapter Review on Rates of Change
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Solutions 18 Videos

The following table shows the daily number of watches sold at a shop and the amount of money made from the sales.

a) Does the data in the table appear to follow a linear relation? Explain.

b) What is the average rate of change in revenue from w = 20 to w = 25?

c) What is the cost of one watch, and how does this cost relate to the graph?

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4.41mins
Q1

The graph shows the height above the ground of a person riding a Ferris wheel.

a) Calculate the average rate of change in height on the interval [0, 4].

b) Calculate the average rate of change in height on the interval [4, 8].

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1.22mins
Q2

A company is opening a new office. The initial expense to set up the office is \$10 000, and the company will spend another \$2500 each month in utilities until the new office opens.

a) Write the equation that represents the company’s total expenses in terms of months until the office opens.

b) What is the average rate of change in the company’s expenses from 3 \leq m \leq 6?

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0.52mins
Q3

In investments value, V( t), is modelled by the function V(t) = 2500(1.15)^t, where t is the number of years after funds are invested.

Find the instantaneous rate of change in the value of the investment at t = 4.

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2.04mins
Q4

The height, in centimetres, of a piston attached to a turning wheel at time t, in seconds, is modelled by the equation y = 2 \sin (120^ot).

Find the instantaneous rate of change at t = 12 s.

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2.37mins
Q5

For the graph shown, estimate the slope of the tangent line at (4, 2).

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1.35mins
Q6a

For the graph shown, estimate the slope of the tangent line at (5, 1).

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0.15mins
Q6b

For the graph shown, estimate the slope of the tangent line at (7, 5).

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0.39mins
Q6c

A sculptor makes a vase for flowers. The radius and circumference of the vase increase as the height of the vase increases. The vase is filled with water. Draw a possible graph of the height of the water as time increases.

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1.03mins
Q8

A newspaper carrier delivers papers on her bicycle. She bikes to the first neighbourhood at a rate of 10 km/h. She slows down at a constant rate over a period of7 s, to a speed of 5 km/h, so that she can deliver her papers. After travelling at this rate for 3 5, she sees one of her customers and decides to stop. She slows at a constant rate until she stops. It takes her 6 s to stop.

a) What is the average rate of change in speed over the first 7 s?

b) What is the average rate of change in speed from second 7 to 12 seconds.

c) What is the instantaneous rate of change in speed at 12 s?

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3.50mins
Q9

The graph shows the height of a roller coaster versus time. Describe how the vertical speed of the roller coaster will vary as it travels along the track from A to G. Sketch a graph to show the vertical speed of the roller coaster.

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2.23mins
Q10

A maximum or minimum is given for each of the following functions. Select a strategy, and verify whether the point given is a maximum or a minimum.

f(x) =x^2 -10x + 7; (5, -18)

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0.21mins
Q11a

A maximum or minimum is given for each of the following functions. Select a strategy, and verify whether the point given is a maximum or a minimum.

g(x) = -x^2 -6x - 4; (-3, 5)

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0.19mins
Q11b

A maximum or minimum is given for each of the following functions. Select a strategy, and verify whether the point given is a maximum or a minimum.

h(x) = -2x^2 + 68x + 75; (17, 653)

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0.10mins
Q11c

A maximum or minimum is given for each of the following functions. Select a strategy, and verify whether the point given is a maximum or a minimum.

j(x) = \sin(-2x); (45^o, -1)

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0.28mins
Q11d

A maximum or minimum is given for each of the following functions. Select a strategy, and verify whether the point given is a maximum or a minimum.

k(x) = -4\cos(x + 25); (-25^o, -4)

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0.45mins
Q11e

A maximum or minimum is given for each of the following functions. Select a strategy, and verify whether the point given is a maximum or a minimum.

m(x) = \frac{1}{20}(x^3 + 2x^2 -15x); (-3, \frac{9}{5})

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0.59mins
Q11f

a) For f(x), find the equation for the slope of the secant line between any general point on the function (a + h, f(a + h)) and the given x-coordinate of another point.

  • f(x) = x^2 -30x; a = 2

b) Use each slope equation you found in part a) to estimate the slope of the tangent line at the point with the given x—coordinate.

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2.03mins
Q12i