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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

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

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0.28mins

Q11d

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

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0.45mins

Q11e

`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