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?
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].
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?
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.
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.
For the graph shown, estimate the slope of the tangent line at (4, 2).
For the graph shown, estimate the slope of the tangent line at (5, 1).
For the graph shown, estimate the slope of the tangent line at (7, 5).
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.
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?
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.
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)
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)
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)
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)
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)
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})
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.