Grade by Grade
D. Characteristics of Functions · Rates of Change

Rates of Change: Unit Review

D1.1 D1.2 D1.3 D1.4 D1.5 D1.6 D1.7 D1.8 D1.9About 40 minutes
Learning goals
  • I can calculate and interpret average rates of change from equations, tables and graphs.
  • I can estimate and calculate instantaneous rates of change with tangents, shrinking intervals, secants and the difference quotient.
  • I can describe rates of change for polynomial, exponential, logarithmic, sinusoidal and rational functions and solve problems with them.

Words to know

average rate of change
(f(b) − f(a)) ÷ (b − a): the slope of the secant over [a, b].
instantaneous rate of change
The rate at one point: the slope of the tangent there.
difference quotient
(f(a + h) − f(a)) ÷ h. Letting h approach 0 gives the tangent slope.

Let's review!

In this unit you described rates of change in words, tables, graphs and equations and sorted them into zero, constant and changing. You sketched graphs from stories. You found average rates as slopes of secants, and instantaneous rates as slopes of tangents, using three tools: shrinking intervals, secants approaching a tangent, and the difference quotient. You then explored the rates of exponential, logarithmic, sinusoidal and rational functions, read rates from graphs, and solved multi-step problems.

This review has a few questions from every lesson. Try each one, then press Show answer to check.

How to use this review

Do two or three sections at a time with a calculator in radian mode (for the sinusoidal questions). For every rate, write the units and say whether it is average or instantaneous. When a question gives an equation, try two methods (for example, a centred interval and the difference quotient) and check that they agree. For each question you miss, note why and go back to that lesson. Finish with the 10-question quiz.

Describing rateswords, tables, sketches (1, 2)Average rateslope of a secant (3, 4)Instantaneous rateslope of a tangent (5)Shrink the intervalnumbers and tables (6)Secants → tangentgraphs and slopes (7)Difference quotientexact slopes (8)Exponential, logrates ∝ value (9)Sinusoidal, rational(10, 11)Graphs and problemsslope graphs (12, 13)
How the unit fits together: describe rates, then average rates (secants) and instantaneous rates (tangents), three ways to find tangent slopes, and the rates of each function family.
Just for fun

You have now spent a whole unit sneaking up on a single point with smaller and smaller intervals. In MCV4U this sneaking up gets an official name, the limit, and its own notation. Same idea, fancier outfit.

Rates of Change All Around Us

1

The temperature rose from −6 °C at 7 a.m. to 4 °C at noon. Find the average rate of change.

2

Classify the rate of change of y = −2x + 3, y = x³ and y = 7.

3

Rent rises 5% every year. Is the rate of change of rent, in dollars per year, constant? Explain.

Sketching Graphs from Descriptions of Rates of Change

1

Describe the graph of the temperature of a hot drink left on a counter.

2

Water is poured at a constant rate into a cone with its point at the bottom. Describe the height-time graph.

3

A motion graph falls in a straight line from d = 4 m to d = 1 m over 3 s. Describe the motion.

Average Rate of Change: The Slope of a Secant

1

Find the average rate of change of f(x) = x² + x on [1, 4].

2

Write the equation of the secant line through the points of f(x) = x² + x at x = 1 and x = 4.

3

Find the average rate of change of g(x) = 2ˣ on [1, 4].

Average Rates of Change in Applications

1

A culture grows as N(t) = 200(3)^(t/2) cells after t hours. Find the average rate of change over [0, 4].

2

For the tide d(t) = 3 cos(πt/6) + 5, find the average rate of change over [3, 9]. Interpret.

3

A table gives 10 cases on day 0, 30 on day 4 and 44 on day 6. Compare the average rates on [0, 4] and [4, 6].

Instantaneous Rate of Change: The Slope of a Tangent

1

A tangent at P passes through (1, 3) and (4, 12). What is the instantaneous rate of change at P?

2

Average or instantaneous: "Our average speed on the commute was 42 km/h."

3

What is the slope of the tangent at the lowest point of a smooth U-shaped graph?

Approximating Instantaneous Rates by Shrinking the Interval

1

Estimate the instantaneous rate of change of f(x) = x² at x = 4 using a centred interval with h = 0.1.

2

Repeat with a following interval, h = 0.01.

3

A table gives T = 80 °C at t = 0, 62 °C at t = 5 and 49 °C at t = 10 min. Estimate the rate at t = 5.

Secants Approaching a Tangent

1

For y = x² and P(3, 9), find the slopes of PQ for Q at x = 3.1 and x = 2.9.

2

From the previous question, state the tangent slope at P and the tangent's equation.

3

The tangent to y = 2ˣ at (0, 1) has slope about 0.693. Write its equation.

The Difference Quotient: Tangent Slopes Algebraically

1

Use the difference quotient to find the tangent slope of f(x) = x² − 5x at x = 1.

2

Find the tangent slope of f(x) = 2x³ at x = 1.

3

A ball's height is h(t) = −5t² + 30t. Find its velocity at time t = a, and at t = 2.

Rates of Change of Exponential and Logarithmic Functions

1

Estimate the tangent slope of y = 2ˣ at x = 4.

2

Estimate the tangent slope of y = log x at x = 20.

3

A substance decays as A(t) = 800(0.5)^(t/10). How does its rate of decay at t = 10 compare with the rate at t = 0?

Rates of Change of Sinusoidal Functions

1

With x in radians, what is the tangent slope of y = sin x at x = π/3?

2

What is the greatest rate of change of y = 4 sin(3x) + 2?

3

On a Ferris wheel, where is the seat's vertical rate of change zero?

Rates of Change of Rational Functions

1

Find the tangent slope of y = 1/x at x = 5.

2

Find the tangent slope of y = 2/x at x = 1.

3

How do the tangent slopes of y = 1/x behave as x gets close to 0 from the right?

Reading Rates of Change from Graphs

1

Describe the slope graph of y = x² − 4x.

2

A slope graph changes from negative to positive at x = −1. What does the function have there?

3

For f(x) = x³ − 12x, estimate the slope at x = 2 with a centred interval (h = 0.001). What does it tell you?

Solving Problems with Rates of Change

1

Find the rate of change of a circle's area with respect to radius at r = 4 cm.

2

For y = x² on [0, 6], where is the tangent parallel to the secant?

3

A town grows as P(t) = 5000(1.02)^t. Estimate its growth rate now.

Check your understanding

Quiz

1.

What is the average rate of change of f(x) = x³ over [0, 2]?

2.

The instantaneous rate of change of f at x = a equals:

3.

Which expression is the centred-interval estimate of the rate of change at x = a?

4.

What is the exact tangent slope of y = x² at x = 5?

5.

For y = 2ˣ, how is the tangent slope at a point related to the y-value there?

6.

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

7.

What is the tangent slope of y = 1/x at x = 3?

8.

Which function has a rate of change of 0 at x = 0 but no maximum or minimum there?

9.

A hot drink cools quickly at first and then more slowly. Its temperature graph is:

10.

In a table with equal x-steps, all first differences are 4. What kind of rate of change does the relation have?

For parents and teachers

Use the review to find what to revisit:

  • Sign or units errors, or confusing zero/constant/changing rates → Lesson 1 (Rates of Change All Around Us).
  • Mixing up "increasing at a decreasing rate" and similar phrases, or container and motion graphs → Lesson 2 (Sketching Graphs from Descriptions of Rates of Change).
  • Wrong average-rate formula or secant equation → Lesson 3 (Average Rate of Change: The Slope of a Secant).
  • Trouble evaluating exponential or sinusoidal models, or comparing unequal intervals → Lesson 4 (Average Rates of Change in Applications).
  • Confusing average and instantaneous, or using curve points to find a tangent's slope → Lesson 5 (Instantaneous Rate of Change: The Slope of a Tangent).
  • Errors with following, preceding and centred intervals → Lesson 6 (Approximating Instantaneous Rates by Shrinking the Interval).
  • Secant slopes from the left side, or tangent equations → Lesson 7 (Secants Approaching a Tangent).
  • Expansion and cancelling errors in the difference quotient → Lesson 8 (The Difference Quotient: Tangent Slopes Algebraically).
  • Thinking the slope of 2ˣ grows with x instead of with 2ˣ, or log rates → Lesson 9 (Rates of Change of Exponential and Logarithmic Functions).
  • Degree mode, or the greatest rate placed at a peak → Lesson 10 (Rates of Change of Sinusoidal Functions).
  • Fraction algebra in the difference quotient, or behaviour near asymptotes → Lesson 11 (Rates of Change of Rational Functions).
  • Placing zeros of the slope graph in the wrong places → Lesson 12 (Reading Rates of Change from Graphs).
  • Choosing a method or multi-step problems → Lesson 13 (Solving Problems with Rates of Change).

Looking ahead: the next unit, Combining Functions and Modelling, finishes strand D of MHF4U. In Grade 12 Calculus and Vectors (MCV4U), the first unit, Limits and Rates of Change, picks up exactly where this unit ends: the "let h approach 0" idea becomes the limit, and the slope graph becomes the derivative.