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Question 1200529: Consider the function f(x) = x^2 + 9
a. Demonstrate how to find the average rate of change from x= -3 to x= 1.
b. Demonstrate algebraically how to find the simplification of f(a+h)-f(a)/h for the given f(x).
c. Let -3 = a, and 1 = a+h, find h. Put that into the simplification in part b. Compare it to the answer for part a. What do you notice?
Answer by GingerAle(43)   (Show Source):
You can put this solution on YOUR website! **a. Find the Average Rate of Change**
* **Calculate f(-3):**
f(-3) = (-3)² + 9 = 9 + 9 = 18
* **Calculate f(1):**
f(1) = (1)² + 9 = 1 + 9 = 10
* **Calculate the Average Rate of Change:**
Average Rate of Change = (f(1) - f(-3)) / (1 - (-3))
= (10 - 18) / (1 + 3)
= -8 / 4
= -2
**Therefore, the average rate of change of f(x) from x = -3 to x = 1 is -2.**
**b. Simplify f(a+h) - f(a) / h**
1. **Find f(a+h):**
f(a+h) = (a+h)² + 9
= a² + 2ah + h² + 9
2. **Find f(a):**
f(a) = a² + 9
3. **Substitute and Simplify:**
(f(a+h) - f(a)) / h
= [(a² + 2ah + h² + 9) - (a² + 9)] / h
= (a² + 2ah + h² + 9 - a² - 9) / h
= (2ah + h²) / h
= 2a + h
**Therefore, (f(a+h) - f(a)) / h simplifies to 2a + h.**
**c. Find h and Substitute**
* Given:
* -3 = a
* 1 = a + h
* Find h:
1 = -3 + h
h = 4
* Substitute h = 4 and a = -3 into the simplified expression:
2a + h = 2(-3) + 4 = -6 + 4 = -2
**Observation:**
The result of the simplification in part (c) (-2) is the same as the average rate of change calculated in part (a).
**Interpretation:**
This demonstrates that the simplification of (f(a+h) - f(a)) / h represents the slope of the secant line between the points (a, f(a)) and (a+h, f(a+h)) on the graph of f(x). In this case, it gives the slope of the secant line between the points (-3, f(-3)) and (1, f(1)).
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