Cogito
AP Calculus AB · Chapter 2 · Lesson 1
The Derivative as a Limit
Bringing the two points together.
11 problems · about 25 minutes · AP Calculus AB 2.1, 2.2
What this lesson teaches
The student defines the derivative as the limit of the difference quotient and interprets it as a tangent slope.
- A secant slope is an average rate of change between two points.
- The derivative is the limit of that slope as the points come together.
- It is the tangent slope, and the instantaneous rate of change.
Warm Up
Straightforward practice. Get the method working first.
4 problemsf(x) = x². What is f'(7)?
Answer 14
Why 14.
f(x) = x². What is f'(1)?
Answer 2
Why 2x.
f(x) = x². What is f'(0)?
Answer 0
Why The vertex is flat.
f(x) = 5x + 2. What is f'(9)?
Answer 5
Why A line has one slope.
Build It Up
The same ideas with more to keep track of.
3 problemsSet the two points together at x = 2 so the secant becomes the tangent, and the slope reads 4.
Answer pointX = 2, secondX = 2
Why Both controls end on the same value.
The difference quotient for f(x) = x² at a = 4 simplifies to 8 + h. What is the derivative?
Answer 8
Why Let h go to 0.
f(x) = x³. What is f'(2)? The derivative of x³ is 3x².
Answer 12
Why 3 × 4.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsIs the derivative an average rate or an instantaneous rate? 1 for average, 2 for instantaneous.
Answer 2
Why The two points have met.
f(x) = 7. What is f'(3)?
Answer 0
Why A constant never changes.
From the Definition: For f(x) = x², what is f'(5)?
Answer 10
Why 10.
Reading the Tangent: The tangent to y = x² at x = 2 is shown. What is its slope?
Answer 4
Why 4.