Wemazi

Topic 2.3

Read a tangent slope from a graph

Builds on choosing a left, right or centered estimate from a table. 6 minutes.

A thrown ball and its tangent

The graph shows the height of a ball thrown upward, in meters, against time in seconds.

The orange line touches the curve at the orange dot and has the same direction as the curve there. This line is the tangent, and its slope is the derivative, how fast the height changes at that moment.

At 1 second, the tangent passes through the chalk dots at (0, 5) and (2, 25). The green leg shows the rise of 20 m and the faint leg shows the run of 2 s, so the slope is 10 m/s.

Go to 2 seconds. The tangent is flat there, so the slope is 0. Go to 3 seconds. The tangent falls to the right, so the rise is negative, its leg is red, and the slope is −10 m/s.

Any two points on the tangent

P is the orange point on the curve, and the orange line is the tangent at P.

Drag A and B along the tangent. The rise and the run both change, but the rise divided by the run stays the same, because a straight line has one slope.

This means that any two points on the tangent give its slope. Choose two that are far apart and on grid lines, so that you can read their coordinates accurately.

Notation

  1. Find the tangent, the line that touches the curve at the point and follows its direction.

  2. Choose two points \((t_1, y_1)\) and \((t_2, y_2)\) on the tangent line, where it crosses grid lines. Do not use points on the curve, because they give a secant.

  3. The derivative at the point is about equal to the rise divided by the run between these two points. It is an estimate, because you read the points from a drawing.

    \[f'(t) \approx \cy{\dfrac{y_2-y_1}{t_2-t_1}}\]

    The unit is the unit of \(y\) per unit of \(t\), for example meters per second.

When it fails

The red line also passes through the orange dot, but it meets the curve again at 4 seconds, so it is a secant.

Its slope from (1, 15) to (4, 0) is −5 m/s, which cannot be the rate, because the ball rises at 1 second.

Example 1: A pool that drains

A graph shows the depth of water in a pool, in centimeters, against time in hours. At 2 hours the tangent passes through (0, 10) and (4, 2). Estimate how fast the depth changes at 2 hours.

  1. The tangent falls from (0, 10) to (4, 2), so the rise in depth is negative.

    \[\cr{2-10 = -8}\]

  2. The run is the change in time.

    \[\cq{4-0 = 4}\]

  3. The depth changes at about −2 cm per hour, and the negative sign means that it falls.

    \[\cy{\dfrac{-8}{4} = -2}\]

Example 2: A train speeding up

A graph shows the speed of a train in m/s against time in seconds. The orange dot is at 4 s, where the speed is 24 m/s. The tangent there passes through (0, 12) and (8, 36). Estimate how fast the speed changes at 4 s.

  1. A tempting wrong move: Use the origin and the orange dot, because both are on the graph.

    \[\cr{\dfrac{24-0}{4-0} = 6}\]

    The red line meets the curve at the origin and at the orange dot, so it is a secant. It is twice as steep as the tangent.

  2. Use the two points on the tangent. The rise is 36 − 12 = 24 and the run is 8 − 0 = 8.

    \[\cy{\dfrac{24}{8} = 3}\]

  3. At 4 seconds, the speed increases at about 3 m/s each second.

Practice

A graph shows the pressure of a gas in kPa against its volume in liters. At 1 liter the tangent passes through (0, 8) and (2, 0). Estimate the slope of the graph at 1 liter.

Move A and B to where the tangent crosses the two axes. Then divide the rise by the run.

Solution

  1. The rise is the change in pressure, and the run is the change in volume.

    \[\cr{0-8 = -8}, \qquad \cq{2-0 = 2}\]

  2. At 1 liter, the pressure changes at about −4 kPa per liter.

    \[\cy{\dfrac{-8}{2} = -4}\]

Questions

Questions 1 and 2 use the graph of the height of the ball and its orange tangent. Question 3 needs a calculation.

  1. The orange dot is at the top of the path, at 2 seconds. What is the slope of the tangent?

    • 0
    • 20
    • 10
    • It does not exist.
    Answer

    0

  2. The orange dot is at 3 seconds. The tangent passes through (2, 25) and (4, 5). What is its slope?

    • −10 m/s
    • 10 m/s
    • 15 m/s
    • −5 m/s
    Answer

    −10 m/s

  3. A tangent crosses the vertical axis at (0, 6) and the horizontal axis at (4, 0). What is its slope?

    • −1.5
    • 1.5
    • −0.67
    • 6
    Answer

    −1.5

Summary

You choose two points on the tangent line and divide the rise by the run between them.

This works because a straight line has one slope, and the slope of the tangent is the derivative.

Use it when a graph is given and you need the rate at one point.

This is one of four skills in topic 2.3 of Which & Why: AP Calculus.