1. Difference Quotient

2. Consider the function y=3x
What is its slope?
Does the slope ever change?

3. Consider the function y=5 𝑥 2
What is its slope?
Does the slope ever change?

6. The Difference Quotient
The difference quotient finds the rate of change for a function in a small neighborhood.
It’s defined as 𝑓 𝑥+ℎ −𝑓(𝑥) ℎ , or sometimes as 𝑓 𝑥+𝛥𝑥 −𝑓(𝑥) 𝛥𝑥

7. Calculate the difference quotient for the following function
𝑓 𝑥 =10𝑥+2

8. Calculate the difference quotient for the following function
𝑓 𝑥 = 𝑥 2

9. Calculate the difference quotient for the following function
𝑓 𝑥 = 𝑒 𝑥

10. Instantaneous rate of change
lim ℎ→0 𝑓 𝑥+ℎ −𝑓(𝑥) ℎ

11. Find the instantaneous rate of change for the following functions
𝑓 𝑥 = 𝑥 2
𝑔 𝑥 = 𝑥 3
ℎ 𝑥 = 1 𝑥
𝑓 𝑥 = 𝑒 𝑥