2.1: The Derivative and Tangent Line Problem
Graph of the derivative and the tangent line.
https://www.desmos.com/calculator/jsufzg2ctl
Why find the tangent line or instantaneous slope?
• Makes it easier to draw graphs and analyze them
• If we have a function that tells us position of an object over time,
the instantaneous slope will tell us the object's velocity.
• If we have a function that tells us velocity of an object over time,
the instantaneous slope will tell us the object's acceleration.
• Help find minimum and maximum costs for businesses
(optimization)
• Too many more to list here…
Problem:
To find tangent line to a point on a curve,
we need to find the instantaneous slope at a point.
This graph shows a secant line. How can we make it a tangent line?
Definition of a Tangent Line with Slope m
Example 1: Find the slope of the tangent line to the graph of the function at the
given point.
Calculus Chapter 2-Derivatives Page 1
Example 2: Find the slope of the tangent line to the graph of the function at the
given point.
What is the Derivative?
"the derivative of f with respect to x"
All these symbols mean derivative
Derivative
• Finds instantaneous rate of change
• Process of finding derivative is called differentiation
• If a function is differentiable at a point that means it is possible to find the
derivative at that point.
Example 3: Find the derivative of the function by the limit definition.
Calculus Chapter 2-Derivatives Page 2
Example 4: Find the derivative of the function by the limit definition.
Example 5: Sketch the graph of f'. Explain how you found your answer.
Example 6: Find the slope of the tangent line to the graph of the function at the
given point. @ (0, 0)
Calculus Chapter 2-Derivatives Page 3
Example 6: Find the slope of the tangent line to the graph of the function at the
given point. @ (0, 0)
We need another definition of a limit to do this one…
First Definition
Alternate Definition
Example 6 (2nd Try)
Find the slope of the tangent line to the graph of the function at the given point.
@ (0, 0)
Calculus Chapter 2-Derivatives Page 4
Example 7: Use the alternate form of the derivative to find the derivative at x = c
(if it exists).
Example 8: Use the alternate form of the derivative to find the derivative at x = c
(if it exists).
Example 9: Use the alternate form of the derivative to find the derivative at x = c
(if it exists).
Calculus Chapter 2-Derivatives Page 5
Some Fun Facts about Differentiability
• If a function is differentiable at a point, then it is continuous at that point.
(in other words if you can find the derivative and evaluate it, then it is
continuous at that point as well)
• It is possible for a function to be continuous, but not differentiable.
(see example 9)
Example 10: Describe the x-values at which f is differentiable.
Calculus Chapter 2-Derivatives Page 6