D. Characteristics of Functions · Rates of Change

Rates of Change of Rational Functions

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

Learning goals

I can find exact tangent slopes of simple rational functions using the difference quotient and fraction algebra.
I can estimate instantaneous rates of change of rational models numerically.
I can describe how the rate of change behaves near a vertical asymptote and far from it.
I can interpret rates of change in rational models such as average cost and drug concentration.

Words to know

rational function
A function that is a quotient of two polynomials, such as 1/x or (2x + 1)/(x − 1).

vertical asymptote
A vertical line x = a that the graph approaches but never touches, where the denominator is 0.

common denominator
A shared denominator used to subtract fractions, such as 2(2 + h) for 1/(2 + h) − 1/2.

average cost
Total cost divided by the number of items. Often a rational function of the number of items.

Splitting the bill

Your club orders custom hoodies. There is a $300 setup fee, plus $25 per hoodie. If n people order, the cost per hoodie is A(n) = 25 + 300/n. With 10 people it is $55 each; with 30 people, $35 each. Each extra person lowers the price, but by less and less.

That is a rational function, like the ones in Rational Functions and Equations. In this lesson you will find their rates of change, exactly with algebra and approximately with numbers, and see what happens near an asymptote.

Worked example

Example 1: y = 1/x at x = 2, exactly

Use the difference quotient to find the slope of the tangent to f(x) = 1/x at x = 2.

Difference quotient: (1/(2 + h) − 1/2) ÷ h.
Subtract the fractions with the common denominator 2(2 + h): 1/(2 + h) − 1/2 = (2 − (2 + h)) ÷ (2(2 + h)) = −h ÷ (2(2 + h)).
Divide by h (multiply by 1/h): −h ÷ (2(2 + h)h) = −1 ÷ (2(2 + h)).
Let h approach 0: −1 ÷ (2 × 2) = −1/4.

The tangent slope at x = 2 is exactly −1/4. (The centred-interval estimate in Approximating Instantaneous Rates by Shrinking the Interval gave −0.2500 ✓.)

Worked example

Example 2: y = 1/x at any point

Find the tangent slope of f(x) = 1/x at x = a (a ≠ 0). List the slopes at a = 0.5, 1, 2 and 4.

(1/(a + h) − 1/a) ÷ h = ((a − (a + h)) ÷ (a(a + h))) ÷ h = −h ÷ (a(a + h)h) = −1 ÷ (a(a + h)).
Let h → 0: −1 ÷ a².
a = 0.5: −1 ÷ 0.25 = −4. a = 1: −1. a = 2: −0.25. a = 4: −0.0625.

Slope of y = 1/x at x = a is −1/a². It is always negative, very steep near the asymptote x = 0, and close to 0 for large x.

Tangents to y = 1/x at x = 0.5, 1 and 2. Close to the vertical asymptote the graph is steep (slope −4); farther away it flattens (slope −0.25).

Near and far from an asymptote

For rational functions, the rate of change tells you a lot about the shape:

Near a vertical asymptote, the function shoots off toward ±∞, so the rate of change is very large in size. For 1/x, the slope at x = 0.1 is −100.
Far from the asymptote, where the graph approaches a horizontal asymptote, the rate of change approaches 0. For 1/x, the slope at x = 10 is −0.01.

The algebra in the next example uses the same idea: combine fractions, cancel h, then let h approach 0.

Worked example

Example 3: A linear-over-linear function

Find the tangent slope of f(x) = (2x + 1)/(x − 1) at x = 3, and the equation of the tangent.

f(3) = 7/2. f(3 + h) = (7 + 2h)/(2 + h).
f(3 + h) − f(3) = (7 + 2h)/(2 + h) − 7/2 = (2(7 + 2h) − 7(2 + h)) ÷ (2(2 + h)) = (14 + 4h − 14 − 7h) ÷ (2(2 + h)) = −3h ÷ (2(2 + h)).
Divide by h: −3 ÷ (2(2 + h)). Let h → 0: −3/4.
Tangent: y − 7/2 = −(3/4)(x − 3), so y = −0.75x + 5.75.

Slope −3/4; tangent y = −0.75x + 5.75.

Worked example

Example 4: Hoodie prices

For A(n) = 25 + 300/n dollars per hoodie, find (a) the average rate of change from n = 10 to n = 30, and (b) the instantaneous rate at n = 20. Interpret.

(a) A(10) = 55 and A(30) = 35. (35 − 55) ÷ (30 − 10) = −20 ÷ 20 = −1.
(b) The 25 does not change, so only 300/n matters. Its slope is 300 × (−1/n²) = −300/n². At n = 20: −300/400 = −0.75.

(a) −$1.00 per extra person, on average, from 10 to 30 people. (b) At 20 people, each extra person lowers the price by about $0.75. (Strictly, n is a whole number, so this is the rate of the smooth model.)

Worked example

Example 5: A medicine in the blood

The concentration of a medicine in the blood is modelled by C(t) = 5t/(t² + 1) mg/L, t hours after it is taken. Estimate the rate of change at t = 0.5, 1, 2 and 3 h, and interpret.

Use centred intervals with h = 0.001.
t = 0.5: ≈ 2.4 mg/L per hour. t = 1: ≈ 0. t = 2: ≈ −0.6. t = 3: ≈ −0.4.

The concentration rises quickly at first (2.4 mg/L per hour at 30 min), peaks at t = 1 h where the rate is 0 (C = 2.5 mg/L), then falls more and more slowly as the body clears the medicine.

The concentration model with tangents at t = 0.5, 1 and 3 h. The horizontal tangent at t = 1 marks the peak concentration.

Just for fun

Splitting a pizza among more friends: your share keeps shrinking, but each new friend costs you less than the last one did. Mathematically, inviting the whole class is fine. Your stomach may disagree.

Hands-on activity

The pizza split

You need: paper, a pencil and a calculator. (A real pizza is optional but encouraged.)

A large pizza costs $24. If n people share it equally, each pays P(n) = 24/n dollars. Make a table for n = 1 to 8.
Find the change in each person's cost when one more person joins (from n to n + 1). These are average rates of change in dollars per person.
Plot P(n) and describe the shape.
Use −24/n² to find the instantaneous rate of the smooth model at n = 4. Compare with your average rates from 3 to 4 and from 4 to 5.

What to look for: the changes are −12, −4, −2, −1.20, −0.80, about −0.57 and about −0.43 dollars per person: always negative and shrinking in size. At n = 4 the model gives −24/16 = −1.50, between −2 (from 3 to 4) and −1.20 (from 4 to 5). The graph is decreasing at a decreasing rate, like the right half of y = 1/x.

Your turn

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

Use the slope formula −1/a² to find the tangent slope of y = 1/x at x = 4.

Find the tangent slope of y = 1/x at x = −2. Is it positive or negative?

Find the tangent slope of f(x) = 3/x at x = 1.

For A(n) = 25 + 300/n, find the average rate of change from n = 20 to n = 60.

For C(t) = 5t/(t² + 1), estimate the rate of change at t = 2 h with a centred interval, h = 0.001.

When does the concentration C(t) reach its maximum, and what is it?

More practice: warm-up

Find the average rate of change of y = 1/x over [1, 2].

Find the average rate of change of y = 1/x over [2, 4].

For x > 0, are the tangent slopes of y = 1/x positive or negative?

Simplify 1/(2 + h) − 1/2 as a single fraction.

Find the average rate of change of y = 10/x over [1, 5].

More practice: on your own

Use the difference quotient to find the tangent slope of f(x) = 1/(x − 1) at x = 3.

Find the tangent slope of f(x) = (2x + 1)/(x − 1) at x = 2. Compare with the slope at x = 3 (−3/4) and explain the difference.

Estimate the tangent slope of f(x) = 1/x² at x = 1 with a centred interval, h = 0.001.

For a gas at constant temperature, pressure P = 600/V kPa when the volume is V litres. Find the rate of change of pressure at V = 3 L and interpret it.

Compare the tangent slopes of y = 1/x at x = 0.1 and x = 10. What does this show?

Challenge

Find all points on y = 1/x where the tangent slope is −4, and write the tangent at each.

The tangent to y = 1/x at (a, 1/a), a > 0, makes a triangle with the positive x- and y-axes. Show that the area of this triangle is always 2.

For hoodie prices A(n) = 25 + 300/n, find the number of people at which the price is dropping at only 3 cents per extra person.

Check your understanding

Quiz

What is the tangent slope of y = 1/x at x = 2?

For x > 0, what happens to the tangent slopes of y = 1/x as x increases?

For A(n) = 25 + 300/n, A(10) = 55 and A(30) = 35. What is the average rate of change from n = 10 to n = 30?

The concentration C(t) = 5t/(t² + 1) has an instantaneous rate of change of 0 at t = 1. What does this tell you?

How do the tangent slopes of a rational function usually behave very close to a vertical asymptote?

What does (1/(a + h) − 1/a) ÷ h simplify to?

For parents and teachers

Big ideas (D1.4, D1.6, D1.8): rational functions have rates of change that become very large near vertical asymptotes and approach 0 near horizontal asymptotes. For 1/x the tangent slope is −1/a². The algebra uses fraction skills: a common denominator, then cancel h.

How to help: Real rational models are everywhere: cost per person when splitting a bill, average cost per item, time to finish a job when more people help. Ask your teen: "Does adding one more person always help by the same amount?"

Common difficulties:

Subtracting fractions incorrectly: 1/(2 + h) − 1/2 is not 1/h.
Dividing by h by putting h on top instead of in the denominator.
Expecting positive slopes for 1/x on the negative branch (they are negative there too).
Using a centred interval that crosses the asymptote: the estimate is meaningless. Keep a − h and a + h on the same side.