Average Rate To Instantaneous Rate of Change

Aim: How do we use “Average Rate of Change” to estimate “Instantaneous Rate of Change”? 2
f (x + h) − f (x)
2. a. Given a curve f (x) = x + x , find the h
Get Ready: Find , then set h=0. average rate of change from x=1 to x=3. g(x)
=
3
−
2x
a. 2
b. g(x) = x − 3x + 2 b. find the average rate of change from x=1 to x=2 c. find the average rate of change from x=1 to x=1.5 d. find the average rate of change from x=1 to x=1.1 e. find the average rate of change from x=1 to x=1.01 f. Estimate the instantaneous rate of change at 1. Finding Average Rate of Change x=1. 1 2
f (x) = x − 4
g. find the average rate of change from x=1 to 3
Given a curve , find the average x=1+h. After, set h=0. f (b) − f (a)
rate of change ( b − a
) from 0 to 3. Aim: How do we use “Average Rate of Change” to estimate “Instantaneous Rate of Change”? h. Find the average rate of change from x to x+h. d. Find the slope of the secant line during the After, set h=0 and plug in x=1. interval [x, x+h]. After, set h=0 and plug in x=2. 3. Attempt to find the instantaneous rate of 2
change of g(x) = x + 2 at x=2. Use the chart to help you to find the average rate of change over the given interval. 2
4. a. Given the function f (x) = 1 − 2x , find the Interval Average Interval Average average rate of change between the following Rate of Rate of intervals: Change Change [2, 3] [2,1] Interval Average Interval Average [2, 2.5] [2,1.5] Rate of Rate of [2, 2.1] [2, 1.9] Change Change [2, 2.01] [2,1.99] [1, 2] [1,0] [2, 2.001] [2, 1.999] [1, 1.5] [1,.5] [2, [2,1.9999] [1, 1.1] [1, .9] 2.0001] [1, 1.01] [1,.99] a. What is your [1, 1.001] [1, .999] best guess for [1, [1,.9999] the 1.0001] instantaneous b. What is your best rate of change at guess for the x = 2? instantaneous rate of change at x = 1? b. Find the slope of the secant line during the interval [2, 2+h]. Then, set h=0. c. Use that number to write the equation of the tangent line to f(x) at x = 1. d. Use that number to write the equation of the normal line (perpendicular line) to f(x) at x = 1. Aim: How do we use “Average Rate of Change” to estimate “Instantaneous Rate of Change”? 2
e. Find the slope of the secant line during the 6. a. Given f (x) = x + 2x − 5 . Find interval [x, x+h]. After, set h=0 and plug in x=2. f (x + h) − f (x)
h
b. S
et h
=0. c. Plug in x=2 to find the instantaneous rate of change. f (x + h) − f (x)
2
h
5. Given f (x) = 2x . a. Find 2
7. a. Given f (x) = 2x + x +1 . Find b. Set h=0. f (x + h) − f (x)
c. Plug in x=-‐1 to find the instantaneous rate of h
change. b. Set h=0. c. Plug in x=2 to find the instantaneous rate of change. Aim: How do we use “Average Rate of Change” to estimate “Instantaneous Rate of Change”? f (x + h) − f (x)
3
h
8. a. Given f (x) = x . Find b. Set h=0. c. Plug in x=-‐1 to find the instantaneous rate of change.

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