D. Characteristics of Functions · Rates of Change

Rates of Change of Sinusoidal Functions

D1.4 D1.6 D1.8About 70 minutes

Learning goals

I can estimate tangent slopes of sine and cosine functions (in radians) and see the pattern they make.
I can tell where a sinusoidal function changes fastest and where its rate of change is zero.
I can calculate and interpret instantaneous rates in sinusoidal models such as Ferris wheels and tides.
I can explain how amplitude and period affect the size of the rate of change.

Words to know

radian
The angle unit used in this unit: π radians = 180°. Calculators must be in radian mode.

midline
The horizontal line halfway between the maximum and minimum of a sinusoidal graph.

amplitude
Half the distance from the maximum to the minimum.

period
The horizontal length of one complete cycle.

Riding the wave

On a Ferris wheel, you rise slowly at the bottom, climb fastest when you pass the height of the axle, slow down near the top, and then do it all in reverse. Your height is a sinusoidal function of time, and your vertical speed keeps changing in a regular pattern.

In Radian Measure and Trigonometric Functions you graphed y = sin x and y = cos x in radians and modelled real cycles. Now you will measure how fast these functions change. Set your calculator to radian mode for this whole lesson.

Worked example

Example 1: Tangent slopes of y = sin x

Estimate the tangent slope of y = sin x at x = 0, π/6, π/3, π/2, 2π/3, 5π/6 and π, using centred intervals with h = 0.001. Do the slopes match any function you know?

At x = 0: (sin 0.001 − sin(−0.001)) ÷ 0.002 ≈ 1.0000.
At π/6: ≈ 0.8660. At π/3: ≈ 0.5000. At π/2: ≈ 0.0000.
At 2π/3: ≈ −0.5000. At 5π/6: ≈ −0.8660. At π: ≈ −1.0000.
These are exactly the values of cos x at the same angles: cos 0 = 1, cos(π/6) = √3/2 ≈ 0.8660, cos(π/3) = 0.5, and so on.

The tangent slope of y = sin x at any x is cos x (in radians). In the same way, you can check that the slope of y = cos x is −sin x.

Tangents to y = sin x. The slope is 1 where the graph crosses its midline going up, 0 at the peak and the trough, and −1 where it crosses going down.

Top: y = sin x with points every π/6. Bottom: the tangent slope at each of those points. The slope points lie exactly on y = cos x (dashed).

Where is a sinusoid changing fastest?

Reading the graphs above:

At a maximum or minimum, the tangent is horizontal, so the rate of change is 0.
Crossing the midline going up, the graph is steepest: the rate is the greatest positive value.
Crossing the midline going down, the rate is the most negative value.

This matches the Ferris wheel: you rise fastest as you pass the height of the axle, and for an instant you are not rising or falling at all at the very top and bottom.

Worked example

Example 2: The Ferris wheel

A Ferris wheel has radius 20 m, its axle is 21 m above the ground, and it turns once every 60 s. A seat's height is h(t) = 21 − 20 cos(πt/30) metres, t seconds after it leaves the bottom. Find its vertical rate of change at t = 0, 5, 15, 30 and 45 s.

Use centred intervals with h = 0.001 (radian mode).
t = 0 (bottom): ≈ 0 m/s.
t = 5: ≈ 1.047 m/s.
t = 15 (axle height, going up): ≈ 2.094 m/s.
t = 30 (top): ≈ 0 m/s. t = 45 (axle height, going down): ≈ −2.094 m/s.

The seat rises fastest, about 2.09 m/s, at t = 15 s, and falls fastest at t = 45 s. Its vertical rate is 0 at the bottom and the top. (Compare with the average rate over [0, 15], which was 20 ÷ 15 ≈ 1.33 m/s.)

Worked example

Example 3: The tide

The tide model d(t) = 3 cos(πt/6) + 5 (metres, t hours after high tide) was used in Average Rates of Change in Applications. Estimate how fast the water is falling at t = 3 h, and compare with the average rate over [2, 4].

Centred interval, h = 0.001: (d(3.001) − d(2.999)) ÷ 0.002 ≈ −1.571 m/h.
Average over [2, 4], from before: −1.5 m/h.

At mid-tide the water is falling at about 1.57 m/h, a little faster than the 1.5 m/h average over [2, 4]: the average over an interval around the steepest point is slightly smaller in size than the steepest rate.

Amplitude and period change the rate

Compare y = sin x with y = a sin(kx):

A vertical stretch by a factor of a multiplies every rise by a, so it multiplies every slope by a.
A horizontal compression by a factor of 1/k squeezes every run by a factor of k, so it multiplies every slope by k.

So the greatest rate of change of y = a sin(kx) + c (or a cos(kx) + c) is a × k (for a, k > 0). For the Ferris wheel, a = 20 and k = π/30, so the greatest vertical rate is 20 × π/30 = 2π/3 ≈ 2.094 m/s, just as Example 2 found.

Just for fun

Why did the sine wave get a summer job at the fair? It already knew all the ups and downs of the Ferris wheel, and exactly when to take a break at the top.

Hands-on activity

Paper Ferris wheel

You need: a sheet of paper, a compass (careful with its sharp point), a protractor, a ruler and a pencil.

Draw a horizontal ground line near the bottom of the page. Draw a circle of radius 8 cm whose bottom just touches the line. Mark the centre.
Starting at the bottom of the circle, mark 12 points every 30° around the circle (use the protractor at the centre).
Measure the height of each point above the ground line, in order around the circle. Record them in a table (step 0 to 12).
Find the change in height for each step. Where are the changes biggest? Where are they smallest?
Compare your heights with h = 8 − 8 cos(30° × step).

What to look for: heights 0, about 1.1, 4, 8, 12, about 14.9, 16, then back down. The changes are smallest near the bottom and top (about 1.1 cm per step) and biggest at the sides (4 cm per step, from 4 to 8 and from 8 to 12): the rate of change is greatest at the level of the centre (the midline).

Your turn

Estimate the tangent slope of y = sin x at x = π/4. What exact value does it match?

Using "slope of cos x is −sin x", give the tangent slopes of y = cos x at x = 0 and at x = π/2.

For the Ferris wheel h(t) = 21 − 20 cos(πt/30), estimate the vertical rate at t = 10 s.

For the tide d(t) = 3 cos(πt/6) + 5, estimate the rate of change at t = 6 h and at t = 9 h. Interpret.

On [0, 2π], where is the tangent slope of y = sin x greatest? Where is it least?

What is the greatest rate of change of y = 5 sin(2x)? Check by estimating the slope at x = 0.

Why must the calculator be in radian mode for these slopes?

More practice: warm-up

What is the rate of change of a sinusoidal function at a maximum point?

What is the tangent slope of y = sin x at x = π?

Which calculator mode do you need for y = sin x with x in radians?

Is y = sin x increasing or decreasing at x = π?

What is the tangent slope of y = sin x at x = 3π/2?

More practice: on your own

A simplified model for the hours of daylight in a southern Ontario city is D(t) = 12.2 + 3.2 sin(2π(t − 80) ÷ 365), t days after January 1. Estimate the rate of change of daylight at t = 80 and at t = 172, in minutes per day.

For the Ferris wheel, find the two times in the first 60 s when the seat is rising at exactly 1 m/s. (Hint: the vertical rate is (2π/3) sin(πt/30).)

A mass on a spring moves as y = 4 cos(2πt) cm (t in seconds). Estimate its velocity at t = 0.25 s and state its greatest speed.

Compare the average rate of change of y = sin x over [0, π/2] with the instantaneous rates at x = 0 and at x = π/2.

For the tide d(t) = 3 cos(πt/6) + 5, when in the first 12 h is the water falling fastest, and how fast?

Challenge

Use the compound angle formula sin(a + h) = sin a cos h + cos a sin h (from Trigonometric Identities and Equations) to show that the difference quotient of sin x at x = a is sin a × (cos h − 1) ÷ h + cos a × (sin h) ÷ h. Then explain why the tangent slope is cos a.

Explain why the greatest rate of change of y = a sin(kx) + c is a × k (a, k > 0), and use it to find the greatest rate of change of y = 3 sin(4x) − 7.

A Ferris wheel seat moves around the circle at a constant speed equal to the circumference divided by the period. Show this speed is 2.094 m/s for our wheel, and explain why the greatest vertical rate equals it.

Check your understanding

Quiz

With x in radians, what is the tangent slope of y = sin x at x = 0?

Where is the rate of change of a sinusoidal function equal to zero?

Where does a sinusoidal function increase fastest?

For the Ferris wheel h(t) = 21 − 20 cos(πt/30), what is the vertical rate of change at t = 15 s?

The tangent slopes of y = sin x at x = 0, π/2, π and 3π/2 are 1, 0, −1 and 0. These match the values of which function?

What is the greatest rate of change of y = 5 sin(2x)?

For parents and teachers

Big ideas (D1.4, D1.6, D1.8): for sinusoidal functions, the rate of change itself goes up and down in a cycle. It is zero at the peaks and troughs and largest at the midline. In radians, the slope of sin x is cos x, and the greatest rate of y = a sin(kx) + c is a × k.

How to help: On a swing, a Ferris wheel or watching a pendulum clock, ask: "Where are you moving fastest? Where do you stop for an instant?" Check that your teen's calculator is in radian mode before homework.

Common difficulties:

Degree mode: slopes come out tiny (about 0.017 instead of 1).
Thinking the rate is greatest at the top of the wave (it is zero there).
Forgetting that the vertical shift c does not change any rate of change.
Typing cos(πt/30) without brackets, so the calculator computes cos(π) × t ÷ 30.

Looking ahead: in MCV4U, "the slope of sin x is cos x" becomes the rule that the derivative of sin x is cos x.