Grade by Grade
A. Rate of Change · Limits and Rates of Change

Secants, Tangents and Smooth Functions

A1.2About 65 minutes
Learning goals
  • I can explain why the average rate of change over an interval equals the slope of the secant over that interval.
  • I can explain why the instantaneous rate of change at a point of a smooth function equals the slope of the tangent there.
  • I can find the slope and equation of a secant, and read a tangent's slope from a graph.
  • I can explain why the instantaneous rate of change cannot be found at a corner, a cusp, a break or a vertical tangent.

Words to know

secant
A line through two points on a curve. Its slope is the average rate of change between them.
tangent
The line that touches a smooth curve at a point P and has the same direction as the curve there. It is the limiting position of secants PQ as Q approaches P.
smooth function
A function that is continuous (no breaks) and has no corners or cusps over an interval, so it has a tangent at each point.
continuous
A graph you can draw without lifting your pencil (made precise in the lesson Limit Laws and Continuity).
corner
A point where a graph changes direction suddenly, so the secants from the left and right approach different slopes.
local linearity
The property of a smooth curve that it looks more and more like a straight line (its tangent) as you zoom in on a point.

Slopes are rates

The slope of a line tells you how much y changes for each unit of x. That is exactly what a rate of change is. So every rate of change question is secretly a slope question.

For a curve, there are two kinds of slope:

  • The secant through P(a, f(a)) and Q(b, f(b)) has slope (f(b) − f(a)) ÷ (b − a). That is the average rate of change of f over [a, b]. The two formulas are literally the same.
  • The tangent at P has a slope equal to the instantaneous rate of change of f at x = a.
01230246810xyP(1, 1)Q(3, 9)y = x²Secant PQ (dashed)slope = (9 − 1) ÷ (3 − 1) = 4= average rate on [1, 3]Tangent at P (solid)slope = 2= instantaneous rate at x = 1Slide Q toward P and thesecant swings into the tangent.
On y = x², the secant through P(1, 1) and Q(3, 9) has slope 4, the average rate over [1, 3]. The tangent at P has slope 2, the instantaneous rate at x = 1.

What exactly is a tangent?

In the Grade 11 (MCR3U) unit Quadratic Functions you found tangent lines to parabolas as lines that meet the parabola at exactly one point, and in the MHF4U unit Rates of Change you watched secants approach a tangent. For curves in general, "touches at one point" is not a good definition. A tangent to y = x³ at (1, 1) crosses the curve again at (−2, −8) (you will show this in the Challenge).

Here is the definition calculus uses. Fix P on the curve. Choose a second point Q on the curve and draw secant PQ. Now slide Q along the curve toward P, from either side. If the secants settle into one position, that limiting line is the tangent at P.

Another way to see it: zoom in on P. A smooth curve looks straighter and straighter as you zoom, until it looks like a line. That line is the tangent. This is called local linearity.

Smooth functions

The Ontario curriculum describes a smooth function as one that is continuous, with no corners. On a smooth piece of a graph, every point has a tangent, so every point has an instantaneous rate of change.

At some points the secants do not settle into one line:

  • Corner: secants from the left and from the right approach different slopes, like the point of a V.
  • Cusp: a very sharp point where the secant slopes grow without bound, toward +∞ on one side and −∞ on the other.
  • Break (discontinuity): the graph jumps or has a hole, so there is no single point P for the secants to pivot on.
  • Vertical tangent: the secants settle into a vertical line, whose slope is undefined.

At all four, the instantaneous rate of change cannot be determined.

cornery = |x|cuspy = ∛(x²)jumptwo piecesvertical tangenty = ∛xAt x = 0, none of these has a tangent with a defined slope.
Four ways a graph can fail to have a tangent with a slope at x = 0. In the jump, the open dot shows a point not on the graph. The lesson When Does a Function Have No Tangent? Differentiability explores these with limits.
Worked example

Example 1: Secant slope and equation

For f(x) = x³ − 2x, find the average rate of change over [1, 3] and the equation of the secant line.

  1. f(1) = 1 − 2 = −1 and f(3) = 27 − 6 = 21.
  2. Slope = (21 − (−1)) ÷ (3 − 1) = 22 ÷ 2 = 11. This is the average rate of change.
  3. Point-slope form through (1, −1): y − (−1) = 11(x − 1), so y = 11x − 12.
  4. Check with the other point: 11(3) − 12 = 21 ✓.

Average rate 11; secant y = 11x − 12.

Worked example

Example 2: Reading a tangent

The tangent to y = √x at P(4, 2) is drawn on a graph. It passes through (0, 1). Find the instantaneous rate of change of y = √x at x = 4.

  1. The tangent passes through (0, 1) and (4, 2).
  2. Slope = (2 − 1) ÷ (4 − 0) = 1/4.
  3. The tangent's slope is the instantaneous rate at x = 4.

The instantaneous rate of change at x = 4 is 1/4 = 0.25. (In the lesson on first principles you will prove this exactly.)

Worked example

Example 3: Why no rate at a corner (based on a curriculum sample)

The graph of f(x) = |x − 2| + 1 is a V with its corner at P(2, 1). Explain why the instantaneous rate of change of f cannot be determined at P.

  1. Take Q to the right of P: Q(2 + h, 1 + h) with h > 0. Slope of PQ = ((1 + h) − 1) ÷ h = 1 for every h.
  2. Take Q to the left of P: Q(2 − h, 1 + h) with h > 0. Slope of PQ = ((1 + h) − 1) ÷ (−h) = −1 for every h.
  3. As Q approaches P, the secants from the right stay at slope 1, and the secants from the left stay at slope −1. They do not settle into one line.
  4. A tangent at P would need a single slope, so there is no tangent.

Because the secant slopes from the two sides approach different values (1 and −1), there is no tangent at P, so the instantaneous rate of change at P cannot be determined.

Worked example

Example 4: A rate in context with a corner

A simple model treats a bouncing ball's height as a series of arcs that meet the ground at sharp corners. What does the model say about the ball's velocity at the instant it hits the ground?

  1. Just before the bounce the height is decreasing: secant slopes are negative (the ball is moving down).
  2. Just after the bounce the height is increasing: secant slopes are positive (moving up).
  3. At the bounce the graph has a corner, so the secant slopes from the two sides approach different values.

The model gives no single instantaneous velocity at the bounce. In reality the ball squashes for a few milliseconds and the velocity changes very quickly but smoothly; the corner is a simplification. The sibling unit Applications of Derivatives: Motion, Rates and Optimization returns to motion problems.

Just for fun

Zoom in far enough on any smooth curve and it looks straight. That is local linearity, and it is why the Earth looks flat from your backyard: you are standing extremely zoomed in on a very large, nearly smooth sphere.

Hands-on activity

Secants, tangents and a crease

You need: a sheet of grid paper (or printer paper with a hand-drawn 1 cm grid), a dinner plate or round lid, a pencil, a clear ruler, and a second sheet of paper.

  1. Trace part of the plate's edge onto the grid paper to get a smooth curve. Mark a point P on it.
  2. Mark points Q₁, Q₂, Q₃ on the curve, each closer to P. Draw the secants PQ₁, PQ₂, PQ₃ with the ruler.
  3. Lay the ruler so it just touches the curve at P without cutting across it. Draw that line: it is your tangent. Do the secants swing toward it?
  4. Find the slope of each line by counting grid squares (rise ÷ run).
  5. Now fold the second sheet sharply and open it a little so the crease makes a V. Trace the V onto grid paper. Try to place the ruler as a single tangent at the tip.

What to look for: on the plate curve, the secant slopes get closer and closer to the tangent's slope. At the tip of the V, the ruler can rock between two positions (one along each arm) and never settles on one: that is a corner, where no tangent exists.

Your turn

1

Find the slope of the secant to f(x) = x² + 1 between x = 1 and x = 4.

2

Write the equation of that secant.

3

The tangent to a curve at P(2, 5) also passes through (0, 1). What is the instantaneous rate of change at x = 2?

4

Does f(x) = |x| have a tangent at x = 0? Does g(x) = x²?

5

Find the average rate of change of g(x) = x³ over [−1, 1].

6

A hockey puck slides toward the boards, hits them and bounces straight back. In a simple model, its distance-from-the-boards graph has a corner at the hit. What does that tell you about the velocity at that instant?

7

True or false: a tangent line can cross the curve at another point. Give a reason.

More practice: warm-up

1

Find the slope of the line through (1, 3) and (4, 12).

2

How many points of a curve does a secant pass through (at least)?

3

An average rate of change is the slope of which line: secant or tangent?

4

What is the tangent to the line f(x) = 5x − 2 at any point? What is its slope?

5

Name the feature: a graph changes direction suddenly at a point, like the tip of a V.

More practice: on your own

1

For f(x) = x³ − 2x, find the slope of the secant over [−1, 2].

2

Find the equation of the secant to f(x) = 1/x over [1, 4].

3

For f(x) = |x + 3|, is there an instantaneous rate of change at x = −3? At x = 0?

4

The tangent to y = √x at (9, 3) has slope 1/6. Write its equation and find its y-intercept.

5

f(x) = x² for x ≤ 1 and f(x) = 2x − 1 for x > 1. Is the graph smooth at x = 1? (Recall from the MHF4U lesson The Difference Quotient: Tangent Slopes Algebraically that the tangent to y = x² at x = a has slope 2a.)

Challenge

1

The tangent to y = x³ at (1, 1) is y = 3x − 2. Show that it meets the curve again, and find where.

2

f(x) = x² for x < 2 and f(x) = mx + b for x ≥ 2. Find m and b so the graph is smooth at x = 2.

3

For f(x) = ∛x, find the slopes of the secants from P(0, 0) to Q(h, ∛h) for h = 0.001 and h = 0.000 001. What happens, and what does it mean?

Check your understanding

Quiz

1.

The slope of a secant through two points on the graph of f represents:

2.

A smooth function is best described as one that is:

3.

For f(x) = |x − 2| + 1, secants from the right of x = 2 have slope 1 and secants from the left have slope −1. What can you conclude?

4.

What is the slope of the secant to f(x) = x² + 1 over [1, 4]?

5.

At which kind of point does a graph have a tangent line whose slope is undefined?

6.

The tangent to y = √x at (4, 2) passes through (0, 1). What is the instantaneous rate of change at x = 4?

For parents and teachers

Big idea (A1.2): average rate of change = slope of a secant; instantaneous rate of change = slope of a tangent, but only where the function is smooth (continuous with no corners). At corners, cusps, breaks and vertical tangents there is no instantaneous rate.

How to help: Draw any curve and ask your teen to show a secant and a tangent and say what each slope means. Then draw a V shape and ask why it has no tangent at the tip.

Common difficulties:

  • Thinking a tangent can touch the curve only once.
  • Averaging the left and right slopes at a corner to get a "rate" (not allowed).
  • Mixing up "the tangent is vertical" (slope undefined) with "the tangent is horizontal" (slope 0).
  • Arithmetic slips in point-slope form; always check the line with the second point.