From secant to tangent

What is it about?

The secant is the line through two points of the graph: $P(x_0 \mid f(x_0))$ and $Q(x_0 + h \mid f(x_0 + h))$. Its slope is the difference quotient

It describes the average rate of change of $f$ between $x_0$ and $x_0 + h$. In the slope triangle, $h$ is the horizontal side and $\Delta y = f(x_0 + h) - f(x_0)$ the vertical side.

If $Q$ moves closer and closer to $P$ ($h \to 0$), the secant turns into a limiting position: the tangent at $P$. Its slope is the derivative of $f$ at $x_0$ (instantaneous rate of change):

Important: you must not plug in $h = 0$. That would give $\frac{0}{0}$, and you cannot divide by $0$. Instead you look at what $m_s$ approaches as $h$ gets smaller – from the right ($h > 0$) and from the left ($h < 0$).

Example: $f(x) = x^2$ (h-method)

Cancelling $h$ is allowed as long as $h \neq 0$. So $f'(x_0) = 2x_0$, e.g. $f'(1) = 2$. The tangent at $P(1 \mid 1)$ has the equation $y = 2(x - 1) + 1 = 2x - 1$.

When there is no limit: the kink of $|x|$

For $f(x) = |x|$ and $x_0 = 0$ we get $m_s = \frac{|h|}{h}$. For every $h > 0$ this is $1$, for every $h < 0$ it is $-1$. From the right and from the left you get different values. The limit does not exist, and $f$ is not differentiable at $0$. The graph has a kink there, and there is no tangent.

Try it

1. Drag $Q$ towards $P$ or press h → 0. Watch the difference quotient and the table.
2. Choose a negative $h$: then $Q$ lies to the left of $P$. Is the limit the same? Negative h
3. Set $h = 0$. What does the simulation show, and why?
4. Show the tangent and move $P$. Where is the tangent horizontal? Cubic function
5. Switch on Trace the derivative f′ and slowly drag $P$ from left to right. Which function appears for $\sin x$? Sine
6. Choose $f(x) = |x|$ with $x_0 = 0$. Compare the table for positive and for negative $h$. Kink of |x|

Exercises

Exercise 1: Difference quotients

For $f(x) = x^2$ and $x_0 = 1$, compute the difference quotient for $h = 1$, $h = 0.5$ and $h = 0.1$. Work it out by hand first, then compare with the table. Which value do the results approach?

Exercise 2: h-method with negative h

Use the h-method to find the derivative of $f(x) = x^2$ at $x_0 = -2$. Also compute $m_s$ for $h = 1$ and for $h = -1$. What does the sign of $f'(-2)$ mean?

Exercise 3: Horizontal tangents

The graph of $f(x) = 0.25x^3 - x$ has two points with a horizontal tangent. Find them in the simulation and compute them with $f'(x) = 0.75x^2 - 1$.

Exercise 4: Derivative of the sine function

For $f(x) = \sin x$, switch on the derivative graph and drag the point $P$ across the graph. Which well-known function appears?

Exercise 5: Tangent to the exponential function

For $f(x) = e^x$ and $x_0 = 0$, use a calculator to compute the difference quotient for $h = 1$ and $h = 0.1$. Estimate $f'(0)$ and give the equation of the tangent at $P(0 \mid 1)$.

Exercise 6: The kink of $|x|$

For $f(x) = |x|$ at $x_0 = 0$, compute the difference quotient for $h = 0.1$ and for $h = -0.1$. Does the graph have a tangent at $0$?

Ideas for the classroom

• Use: Upper secondary, introduction to calculus (slopes of secants and tangents, average and instantaneous rate of change, derivative function, tangent equation). The h → 0 animation shrinks $h$ logarithmically, so the approach stays evenly visible.
• Table: The table shows the difference quotients for $h = 1;\ 0.1;\ 0.01;\ 0.001$ with the sign of the chosen $h$. It is a good way to prepare the idea of a limit numerically. Let the class guess or compute the values by hand first.
• h = 0: If you set $h = 0$, the difference quotient is undefined ($\frac{0}{0}$); the simulation then shows only the tangent. This is a good moment to discuss the difference between “plugging in $h = 0$” and “the limit as $h \to 0$”.
• Two-sided limit: Compare $h > 0$ and $h < 0$ first for $x^2$ (same limit) and then for $|x|$ (different values). This makes it tangible why differentiability needs more than “the graph has no jump”.
• Common pitfalls:
  • “The tangent touches the graph at only one point” – for the cubic function the tangent often cuts the graph at another point.
  • Secant and tangent get mixed up. In the simulation the secant is a solid line and the tangent is dashed; the legend shows both line styles.
  • When cancelling $h$, students forget that this only works for $h \neq 0$.