1. Calculus I (MAT 145) Dr. Day Friday September 29, 2017
Derivative Shortcuts (Chapter 3)
Sums & Differences, Products & Quotients, and More! ( )
Derivatives of Trig Functions (3.3)
Derivatives of Composite Functions (3.4)
Friday, September 29, 2017
MAT 145

2. Warm up! . Find the derivatives. Use correct notation!
Friday, September 29, 2017
MAT 145

3. Using Derivative Patterns
MAT 145
Suppose s(x), shown below, represents an object’s position as it moves back and forth on a number line, with s measured in centimeters and x in seconds, for x > 0.
Calculate the object’s velocity and acceleration functions.
Is the object moving left or right at time x = 1? Justify.
Determine the object’s velocity and acceleration at time x = 2. Based on those results, describe everything you can about the object’s movement at that instant.
Write an equation for the tangent line to the graph of s at time x = 1.
Friday, September 29, 2017

4. Using Derivative Patterns
Determine the equation for the line tangent to the graph of g at x = 4.
Determine the equation for the line normal to the graph of g at x = 1.
At what points on the graph of g, if any, will a tangent line to the curve be parallel to the line 3x – y = –5?
Friday, September 29, 2017
MAT 145

5. Sums, differences, exponentials, & products of constants and functions
Friday, September 29, 2017
MAT 145

6. Derivatives of Trig Functions
Friday, September 29, 2017
MAT 145

7. Derivatives of Composite Functions
Friday, February 24, 2017
MAT 145

8. Derivatives of Composite Functions (3.4)
Friday, February 24, 2017
MAT 145

9. Here's a function in a function And your job here is to find
THE CHAIN RULE
WORDS BY: JOHN A. CARTER
TUNE: "CLEMENTINE"
Here's a function in a function
And your job here is to find
The derivative of the whole thing
With respect to x inside.
Call the outside f of u
And call the inside u of x.
Differentiate to find df/du
And multiply by du/dx.
Use the chain rule.
Use the chain rule when e'er you find
The derivative of a function compositionally defined.
Friday, February 24, 2017
MAT 145

10. Derivatives of Composite Functions (3.4)
Friday, February 24, 2017
MAT 145

11. Derivatives of Composite Functions
Friday, February 24, 2017
MAT 145

12. Composite Functions
Friday, February 24, 2017
MAT 145

13. Practice Derivative Rules
Friday, February 24, 2017
MAT 145

14. Using Derivative Patterns
For s(t) = cos(2t):
Calculate s’(t) and s’’(t).
Determine an equation for the line tangent to the graph of s when t = π/8.
Determine the two values of t closest to t = 0 that lead to horizontal tangent lines.
Determine the smallest positive value of t for which s’(t) = 1.
If s(t) represents an object’s position on the number line at time t (s in feet, t in minutes), calculate the object’s velocity and acceleration at time t = π/12. Based on those results, describe everything you can about the object’s movement at that instant.
Friday, February 24, 2017
MAT 145