Cogito
Calculus · Chapter 3 · Lesson 1
From Secant to Tangent
The derivative, built rather than asserted.
10 problems · about 22 minutes · CHA-2.B, CHA-2.C
What this lesson teaches
The student defines the derivative as the limit of a difference quotient and interprets it as a slope.
- A secant slope is an average over an interval.
- As the interval shrinks to nothing, the secant becomes the tangent.
- The derivative is that limit, and it is a slope.
Warm-Up Review
From earlier lessons. Loosen up before the new work.
2 problemsReview — The Intermediate Value Theorem: f(1) = −3 and f(4) = 6. Does the theorem guarantee a root between them? 1 or 0.
Answer 1
Why Yes, assuming continuity.
Review — Types of Discontinuity: Left limit 3, right limit 3, function value 8. Which kind? 1, 2 or 3.
Answer 1
Why Removable.
Warm Up
Straightforward practice. Get the method working first.
4 problemsf′(x) = 2x. Slope at x = 7?
Answer 14
Why 14.
What is a derivative?
Answer The limit of a slope between two points as they come together.
Why A limit of slopes.
f′(x) = 2x. Slope at x = 5?
Answer 10
Why 2 × 5.
f′(x) = 2x. Slope at x = −4?
Answer -8
Why 2 × −4.
Build It Up
The same ideas with more to keep track of.
2 problemsf(x) = 3x. What is f′(x)?
Answer 3
Why A line has constant slope.
What does the secant become as the points meet?
Answer The tangent.
Why One point, one line.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
2 problemsThe Slope: f(x) = x², so f′(x) = 2x. What is the slope at x = 3?
Answer 6
Why 6.
The Turning Point: f(x) = x². At what x is the slope zero?
Answer 0
Why x = 0, the vertex.