D. Characteristics of Functions · Rates of ChangeInstantaneous Rate of Change: The Slope of a Tangent
D1.5 D1.7About 65 minutes
Learning goals- I can recognize real situations that involve an instantaneous rate of change.
- I can explain the difference between an average rate over an interval and an instantaneous rate at a point.
- I can explain that the instantaneous rate of change at a point is the slope of the tangent there.
- I can estimate an instantaneous rate from a graph by drawing a tangent and finding its slope.
Words to know
instantaneous rate of change
The rate of change at one particular moment or point, for example the reading on a speedometer.
tangent line
A line that touches a smooth curve at a point P and has the same direction (steepness) as the curve at P.
point of tangency
The point where the tangent touches the curve.
local linearity
When you zoom in far enough on a smooth curve, it looks almost like a straight line: its tangent.
What does the speedometer know?
On a highway trip you might average 85 km/h. But glance at the speedometer and it says 97 km/h right now. A moment later, it says 100 km/h. The speedometer does not wait for the trip to finish. It shows your speed at one instant.
That is an instantaneous rate of change: the rate at a single point, not over an interval. An average rate needs two points. An instantaneous rate needs only one. So how do you find a slope with only one point? That puzzle is what this lesson, and the next few, are about.
Average vs instantaneous
- An average rate of change is measured over an interval [a, b]. It is the slope of a secant through two points.
- An instantaneous rate of change is measured at a point x = a. It is the slope of the tangent at that point.
Key words help. "From… to…", "over", "during", "between" and "per year last year" signal an average. "At", "when", "at the moment", "right now" and "at exactly" signal an instantaneous rate.
Real examples of instantaneous rates:
- the reading on a speedometer, or the velocity of a ball at t = 2 s;
- how fast a balloon's volume grows at the moment its radius is 10 cm;
- the rate of a chemical reaction at the moment the reactants are mixed;
- the extra cost of making the 101st hoodie (economists call this the marginal cost);
- a heart-rate monitor's reading at one moment.
The tangent line
A tangent line at a point P on a smooth curve touches the curve at P and points in the same direction as the curve there.
Here is a powerful way to see it. Zoom in on the curve near P. The more you zoom, the straighter the curve looks. Eventually it looks just like a straight line, and that line is the tangent. This property is called local linearity.
So the steepness of the curve at P is the slope of the tangent at P:
instantaneous rate of change at x = a = slope of the tangent at (a, f(a))
A tangent is not always "one touch"
In the MCR3U unit Quadratic Functions, a tangent to a parabola met it at exactly one point. For other curves, that is not the rule. A tangent matches the curve's direction at P, but it may cross the curve somewhere else. What matters is how it behaves near P.
Worked exampleExample 1: Average or instantaneous?
Decide whether each statement describes an average or an instantaneous rate of change.
(a) "The radar gun clocked the pitch at 142 km/h as it crossed the plate." (b) "Our town's population grew by about 2% last year." (c) "At 3:15 p.m. the snow was falling at 4 cm/h." (d) "The bus covered the 18 km route in 40 min."
- (a) "as it crossed the plate" is one moment: instantaneous.
- (b) "last year" is an interval: average.
- (c) "At 3:15 p.m." is one moment: instantaneous.
- (d) a whole route over 40 min: average (27 km/h).
(a) instantaneous, (b) average, (c) instantaneous, (d) average.
Worked exampleExample 2: Reading a tangent from a graph
A car pulls away from a stop sign. Its distance is d(t) = 1.25t² metres after t seconds. The tangent at P(4, 20) is drawn on the graph below and passes through (2, 0) and (6, 40). Find the car's velocity at t = 4 s.
- The velocity at t = 4 is the slope of the tangent at P.
- Use two points on the tangent (not on the curve): (2, 0) and (6, 40).
- Slope = (40 − 0) ÷ (6 − 2) = 40 ÷ 4 = 10.
At t = 4 s the car is moving at 10 m/s (36 km/h).
The tangent at P(4, 20) on the car's distance-time graph. Pick two points on the tangent that are far apart and easy to read, then find the slope.Drawing a good tangent by hand
Place a ruler so it touches the curve at P and the curve bends away evenly on both sides (or, near a bend-change, crosses smoothly). Draw a long line. Then pick two points on your line that are far apart and easy to read. Points close together give larger errors.
Worked exampleExample 3: Average rates close in on the tangent
For d(t) = 1.25t², find the average velocity over [4, 5], [4, 4.1] and [4, 4.01]. Compare with the 10 m/s from Example 2.
- d(4) = 20, d(5) = 31.25, d(4.1) = 1.25(16.81) = 21.0125, d(4.01) = 1.25(16.0801) = 20.100125.
- [4, 5]: (31.25 − 20) ÷ 1 = 11.25 m/s.
- [4, 4.1]: (21.0125 − 20) ÷ 0.1 = 10.125 m/s.
- [4, 4.01]: (20.100125 − 20) ÷ 0.01 = 10.0125 m/s.
11.25, 10.125, 10.0125 m/s: the shorter the interval, the closer the average velocity gets to the instantaneous velocity of 10 m/s. The next lesson turns this into a method.
Just for funTangent comes from the Latin tangere, "to touch" (the same root as tangible). So when someone "goes off on a tangent", they are touching your topic for exactly one instant and then heading off in a straight line.
Hands-on activityRuler tangents
You need: graph paper (or a printed grid), a ruler, a sharp pencil and a calculator.
- Draw axes with 2 cm for 1 unit on the x-axis and 1 cm for 1 unit on the y-axis. Plot y = x² for x = 0, 0.5, 1, 1.5, 2, 2.5 and 3, and join the points with a smooth curve.
- At P(1, 1), lay your ruler so it just touches the curve and the curve bends away evenly on both sides. Draw a long line.
- Pick two points on your line far apart. Calculate the slope.
- Repeat at (2, 4) and at (3, 9) (extend your axes if needed).
- Compare your slopes with 2, 4 and 6. What pattern do you see?
What to look for: your slopes should be close to 2, 4 and 6 (within about 10% is good for a hand drawing). The pattern is "slope = 2x" for y = x². You will prove this algebraically in the lesson The Difference Quotient: Tangent Slopes Algebraically.
Your turn
1Average or instantaneous? "The speedometer reads 90 km/h."
Instantaneous: it is the speed at one moment.
2Average or instantaneous? "Our road trip averaged 85 km/h."
Average: over the whole trip.
3Average or instantaneous? "The town grew by 300 people per year from 2016 to 2021."
Average: over the interval 2016 to 2021.
4A tangent at P passes through (1, 4) and (5, 12). What is the instantaneous rate of change at P?
Slope = (12 − 4) ÷ (5 − 1) = 2.
5The tangent to y = x² at (2, 4) passes through (1, 0) and (3, 8). Find the instantaneous rate of change at x = 2.
6For d(t) = 1.25t², compute the average velocity over [4, 4.001]. What does it suggest?
d(4.001) = 1.25(16.008001) = 20.01000125. (20.01000125 − 20) ÷ 0.001 = 10.00125 m/s, very close to the instantaneous velocity of 10 m/s.
7On the graph of the height of a ball thrown straight up, where is the instantaneous rate of change of height zero? What does the tangent look like there?
At the highest point. The tangent there is horizontal (slope 0).
More practice: warm-up
1A tangent passes through (0, 3) and (2, 7). What is its slope?
2What is the instantaneous rate of change of y = 5x − 1 at x = 10?
5. The tangent to a line is the line itself.
3A curve has a horizontal tangent at P. What is the instantaneous rate of change at P?
4A graph is falling as it passes through P. Is the slope of the tangent at P positive or negative?
5Which line touches a curve at a point and has the same direction as the curve there: a secant or a tangent?
More practice: on your own
1For d(t) = 1.25t², estimate the velocity at t = 2 s using the average velocities over [2, 2.1] and [1.9, 2].
d(2) = 5, d(2.1) = 5.5125, d(1.9) = 4.5125. [2, 2.1]: 0.5125 ÷ 0.1 = 5.125 m/s. [1.9, 2]: 0.4875 ÷ 0.1 = 4.875 m/s. The true value is between them; their average gives about 5 m/s.
2On a temperature graph, a tangent at 10:00 passes through (8:00, −9 °C) and (12:00, −1 °C). How fast was the temperature changing at 10:00?
(−1 − (−9)) ÷ 4 = 8 ÷ 4 = 2 °C/h (rising).
3The tangent to y = x³ at (1, 1) is y = 3x − 2. Show that it meets the curve again at x = −2. What does this show about tangents?
At x = −2: 3(−2) − 2 = −8 and (−2)³ = −8, so (−2, −8) is on both. A tangent can meet the curve again away from the point of tangency; it only has to match the curve's direction at P.
4The ball h(t) = −4.9t² + 20t + 1.5 has tangent h = 10.2t + 6.4 at t = 1. Find the ball's velocity at t = 1 and check that the tangent passes through the point on the curve.
Velocity = slope = 10.2 m/s. h(1) = 16.6 and 10.2(1) + 6.4 = 16.6 ✓.
5Give two examples of instantaneous rates of change in a hospital, with units.
Model answer: a heart rate shown on a monitor at one moment (beats per minute); the drip rate of an IV line at a given moment (mL/h); the rate a patient's temperature is rising at 2 p.m. (°C/h). Any rate at a single moment, with units, is correct.
Challenge
1In the lesson Average Rate of Change: The Slope of a Secant you showed that the secant slope of y = x² on [a, b] is a + b. Use this to predict the slope of the tangent at x = a. Check your prediction at (1, 1).
As b gets closer and closer to a, the secant slope a + b gets closer to a + a = 2a. At x = 1 this gives 2, matching the tangent slope of 2 at (1, 1) ✓.
2Can the average rate of change over [0, 4] be positive while the instantaneous rate at t = 4 is negative? Explain with an example.
Yes. For the ball h(t) = −4.9t² + 20t + 1.5, the average velocity over [0, 4] is 0.4 m/s (it ends a bit higher than it started), but at t = 4 the ball is falling, so the slope of the tangent there is negative.
3Explain why, for a linear function, the average rate of change over every interval equals the instantaneous rate of change at every point.
The graph is a straight line. Every secant is the line itself, and the tangent at every point is the line itself. So every secant slope and every tangent slope equals the line's slope m.
Quiz
1. Which of these is an instantaneous rate of change?
2. The instantaneous rate of change of f at x = a is the slope of:
3. A tangent is drawn at P. It passes through (2, 3) and (6, 11). What is the instantaneous rate of change at P?
4. At the top of a smooth hill-shaped graph, the instantaneous rate of change is:
5. Which is the best way to approximate an instantaneous rate of change using average rates?
6. Which statement about tangent lines is true?
Big ideas (D1.5, D1.7): an instantaneous rate of change is a rate at one point; it is the slope of the tangent there. Average rates over very short intervals approximate it.
How to help: In the car, ask: "Is the speedometer showing an average or an instantaneous speed? What would our average speed for the trip be?" At home, draw any smooth curve and ask your teen to draw the tangent at a point with a ruler and explain why it is a tangent.
Common difficulties:
- Using points on the curve instead of points on the tangent to find the tangent's slope.
- Picking two points very close together on a hand-drawn tangent (small reading errors become large slope errors).
- Believing a tangent can never cross or meet the curve again.
- Confusing "at" (instantaneous) and "over" (average) in word problems.