Cogito
Precalculus · Chapter 8 · Lesson 3
The Tangent Line Problem
Where calculus actually begins.
12 problems · about 22 minutes · TEKS P.1.A
What this lesson teaches
The student estimates the slope of a tangent line as the limit of secant slopes.
- A tangent slope is the limit of secant slopes as the second point closes in.
- That limit of [f(x + h) − f(x)] ÷ h is the derivative.
- It measures the instantaneous rate of change.
Warm Up
Straightforward practice. Get the method working first.
5 problemsOn y = x², what is the tangent slope at x = 6?
Answer 12
Why 12.
Why must a limit be used rather than simply setting h = 0?
Answer Setting h = 0 gives 0 ÷ 0, which says nothing.
Why It produces the indeterminate form 0 ÷ 0.
On y = x², tangent slope at x = 3?
Answer 6
Why 2x.
On y = x², tangent slope at x = 0?
Answer 0
Why The vertex is flat.
On y = x², tangent slope at x = −4?
Answer -8
Why 2 × −4.
Build It Up
The same ideas with more to keep track of.
3 problemsOn y = x², points at x = 2 and x = 4. Secant slope?
Answer 6
Why (16 − 4) ÷ 2.
On the line y = 3x + 1, what is the tangent slope anywhere?
Answer 3
Why A line is its own tangent.
Setting h = 0 in the difference quotient gives which form? Enter 1 for 0 ÷ 0, 2 for a number.
Answer 1
Why Both parts vanish.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsPut the tangent line process in order.
Answer 1. Choose a second point a distance h away. 2. Write the secant slope as a difference quotient. 3. Simplify until h cancels from the denominator. 4. Take the limit as h approaches zero.
Why The cancelling must happen before the limit can be taken.
On y = x², tangent slope at x = 7?
Answer 14
Why 2 × 7.
The Secant: On y = x², the points at x = 1 and x = 3. What is the secant slope?
Answer 4
Why 4.
The Tangent: On y = x², the tangent slope is 2x. What is it at x = 5?
Answer 10
Why 10.