How does a secant slope become a tangent slope?
Quick answer
A secant slope measures the change between two points on a curve. Fix one point at x=a and move the other to x=a+h. Their slope is [f(a+h)−f(a)]/h, for h≠0. As h approaches 0, this secant slope may approach a limit. If that limit exists, it is the tangent slope—the derivative at a. For f(x)=x² at a=1, the quotient simplifies to 2+h, so the tangent slope is 2. [S1]
The question, the details, the next step.
I know how to find the slope between two points, but how does that give me the slope of a curve at just one point?
Start with two points on the curve. Keep the first at x=a, and put the second at x=a+h. Their vertical change is f(a+h)−f(a), and their horizontal change is h, so the secant slope is [f(a+h)−f(a)]/h, with h≠0. [S1]
Now imagine moving the second point closer to the first by making h approach 0. If the secant slopes approach a limit, that limit is the tangent slope at x=a—the derivative there. The secant uses two distinct points; the tangent slope comes from the limiting behavior of the secants. [S1]
Can you work through that for f(x)=x² at x=1? I get lost when I substitute the nearby point.
The fixed input is 1, and the nearby input is 1+h. Their function values are f(1)=1 and f(1+h)=(1+h)². Substitute them into the secant-slope formula:
[f(1+h)−f(1)]/h = [(1+h)²−1]/h
Expand and simplify, remembering that h is nonzero in the original quotient:
[(1+2h+h²)−1]/h = (2h+h²)/h = 2+h
For any nonzero h, the secant slope is 2+h. As h approaches 0, 2+h approaches 2. Therefore the tangent slope at x=1 is 2. [S1]
Why can’t I just put h=0 into the quotient? And what does h represent on a graph?
Putting h=0 into the original quotient makes its denominator zero, so the quotient is undefined there. Instead, simplify it for h≠0, then find what value the simplified expression approaches as h gets closer to zero. In this example, 2+h approaches 2. The limit describes behavior near h=0; it does not require evaluating the original fraction at h=0.
On a graph, h is the horizontal distance from the fixed input a to the nearby input a+h. You can picture choosing small nonzero values of h and comparing the secant slopes. The limit gives the tangent slope only when it exists; this example does not show that every function has a tangent slope at every point. [S1]
What’s a reliable way to set this up on another function without mixing up the points?
Write the fixed input and moving input first: a and a+h. Then write their outputs in the same order, f(a) and f(a+h), and form (moving output − fixed output)/(moving input − fixed input). That gives [f(a+h)−f(a)]/h. Keep h≠0 while simplifying; only then consider the limit as h approaches 0. If the limit exists, it gives the tangent slope at a. [S1]
Try this template:
f(a)=___
f(a+h)=___
[f(a+h)−f(a)]/h = ___
Simplify for h≠0: ___
Limit as h→0: ___
Tangent slope at x=a, if the limit exists: ___
For the worked example, a=1, f(a)=1, f(a+h)=(1+h)², the simplified quotient is 2+h, and the limit is 2. If the quotient does not approach a limit, this procedure does not give a tangent slope at that point.
A practical checklist
Choose the point of interest and label its input a.
Represent a nearby input as a+h; keep h nonzero in the quotient.
Substitute f(a+h) and f(a) in that order: [f(a+h)−f(a)]/h.
Simplify the quotient before considering h→0.
Report the limit as the tangent slope only if it exists.
A few more useful answers
Is the secant slope itself the tangent slope?
Not usually: a secant slope uses two distinct points. The tangent slope is the limit of those secant slopes as the second point approaches the first, provided the limit exists. [S1]
Why is h required to be nonzero?
In the difference quotient, h is the denominator, so the quotient is undefined at h=0. Simplify for nonzero h, then take the limit as h approaches zero.
Does this method always produce a tangent slope?
Only when the difference quotient has a limit at the point. The example shows a case where it does; it does not establish that the limit exists for every function or point. [S1]
Sources
References used for this guide.
- [S1] OpenStax Calculus Volume 1, Section 3.1: Defining the Derivative openstax.org · Checked