Answer :
Sure! Let's go through each of the exercises one by one to find and compare the average rate of change with the instantaneous rates of change at the endpoints of the given intervals.
### Exercise 91: [tex]\( f(t) = 3t + 5 \)[/tex], Interval: [tex]\([1, 2]\)[/tex]
1. Average Rate of Change:
- Calculate [tex]\( f(2) \)[/tex] and [tex]\( f(1) \)[/tex]:
[tex]\[
f(2) = 3 \cdot 2 + 5 = 11
\][/tex]
[tex]\[
f(1) = 3 \cdot 1 + 5 = 8
\][/tex]
- The average rate of change is:
[tex]\[
\frac{f(2) - f(1)}{2 - 1} = \frac{11 - 8}{1} = 3
\][/tex]
2. Instantaneous Rates of Change:
- The derivative of [tex]\( f(t) = 3t + 5 \)[/tex] is [tex]\( f'(t) = 3 \)[/tex].
- At [tex]\( t = 1 \)[/tex] and [tex]\( t = 2 \)[/tex], the rate is 3.
Comparison: Both the average rate of change and the instantaneous rates at the endpoints are 3.
### Exercise 92: [tex]\( f(t) = t^2 - 7 \)[/tex], Interval: [tex]\([3, 3.1]\)[/tex]
1. Average Rate of Change:
- Calculate [tex]\( f(3.1) \)[/tex] and [tex]\( f(3) \)[/tex]:
[tex]\[
f(3.1) = (3.1)^2 - 7 = 2.61
\][/tex]
[tex]\[
f(3) = 3^2 - 7 = 2
\][/tex]
- The average rate of change is:
[tex]\[
\frac{f(3.1) - f(3)}{3.1 - 3} = \frac{2.61 - 2}{0.1} = 6.1
\][/tex]
2. Instantaneous Rates of Change:
- The derivative of [tex]\( f(t) = t^2 - 7 \)[/tex] is [tex]\( f'(t) = 2t \)[/tex].
- At [tex]\( t = 3 \)[/tex], the rate is [tex]\( 2 \cdot 3 = 6 \)[/tex].
- At [tex]\( t = 3.1 \)[/tex], the rate is [tex]\( 2 \cdot 3.1 = 6.2 \)[/tex].
Comparison: The average rate of change is 6.1, while the instantaneous rates at the endpoints are 6 and 6.2.
### Exercise 93: [tex]\( f(x) = \frac{-1}{x} \)[/tex], Interval: [tex]\([1, 2]\)[/tex]
1. Average Rate of Change:
- Calculate [tex]\( f(2) \)[/tex] and [tex]\( f(1) \)[/tex]:
[tex]\[
f(2) = \frac{-1}{2} = -0.5
\][/tex]
[tex]\[
f(1) = \frac{-1}{1} = -1
\][/tex]
- The average rate of change is:
[tex]\[
\frac{f(2) - f(1)}{2 - 1} = \frac{-0.5 + 1}{1} = 0.5
\][/tex]
2. Instantaneous Rates of Change:
- The derivative of [tex]\( f(x) = \frac{-1}{x} \)[/tex] is [tex]\( f'(x) = \frac{1}{x^2} \)[/tex].
- At [tex]\( x = 1 \)[/tex], the rate is [tex]\( 1 \)[/tex].
- At [tex]\( x = 2 \)[/tex], the rate is [tex]\( \frac{1}{4} \)[/tex].
Comparison: The average rate of change is 0.5, while the instantaneous rates at the endpoints are 1 and 0.25.
### Exercise 94: [tex]\( f(x) = \sin x \)[/tex], Interval: [tex]\([0, \frac{\pi}{6}]\)[/tex]
1. Average Rate of Change:
- Calculate [tex]\( f(\frac{\pi}{6}) \)[/tex] and [tex]\( f(0) \)[/tex]:
[tex]\[
f\left(\frac{\pi}{6}\right) = \sin\left(\frac{\pi}{6}\right) = 0.5
\][/tex]
[tex]\[
f(0) = \sin(0) = 0
\][/tex]
- The average rate of change is:
[tex]\[
\frac{f\left(\frac{\pi}{6}\right) - f(0)}{\frac{\pi}{6} - 0} = \frac{0.5 - 0}{\frac{\pi}{6}} = \frac{3}{\pi} \approx 0.9549
\][/tex]
2. Instantaneous Rates of Change:
- The derivative of [tex]\( f(x) = \sin x \)[/tex] is [tex]\( f'(x) = \cos x \)[/tex].
- At [tex]\( x = 0 \)[/tex], the rate is [tex]\( \cos(0) = 1 \)[/tex].
- At [tex]\( x = \frac{\pi}{6} \)[/tex], the rate is [tex]\( \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} \approx 0.866 \)[/tex].
Comparison: The average rate of change is approximately 0.9549, with instantaneous rates at the endpoints being 1 and approximately 0.866.
### Exercise 91: [tex]\( f(t) = 3t + 5 \)[/tex], Interval: [tex]\([1, 2]\)[/tex]
1. Average Rate of Change:
- Calculate [tex]\( f(2) \)[/tex] and [tex]\( f(1) \)[/tex]:
[tex]\[
f(2) = 3 \cdot 2 + 5 = 11
\][/tex]
[tex]\[
f(1) = 3 \cdot 1 + 5 = 8
\][/tex]
- The average rate of change is:
[tex]\[
\frac{f(2) - f(1)}{2 - 1} = \frac{11 - 8}{1} = 3
\][/tex]
2. Instantaneous Rates of Change:
- The derivative of [tex]\( f(t) = 3t + 5 \)[/tex] is [tex]\( f'(t) = 3 \)[/tex].
- At [tex]\( t = 1 \)[/tex] and [tex]\( t = 2 \)[/tex], the rate is 3.
Comparison: Both the average rate of change and the instantaneous rates at the endpoints are 3.
### Exercise 92: [tex]\( f(t) = t^2 - 7 \)[/tex], Interval: [tex]\([3, 3.1]\)[/tex]
1. Average Rate of Change:
- Calculate [tex]\( f(3.1) \)[/tex] and [tex]\( f(3) \)[/tex]:
[tex]\[
f(3.1) = (3.1)^2 - 7 = 2.61
\][/tex]
[tex]\[
f(3) = 3^2 - 7 = 2
\][/tex]
- The average rate of change is:
[tex]\[
\frac{f(3.1) - f(3)}{3.1 - 3} = \frac{2.61 - 2}{0.1} = 6.1
\][/tex]
2. Instantaneous Rates of Change:
- The derivative of [tex]\( f(t) = t^2 - 7 \)[/tex] is [tex]\( f'(t) = 2t \)[/tex].
- At [tex]\( t = 3 \)[/tex], the rate is [tex]\( 2 \cdot 3 = 6 \)[/tex].
- At [tex]\( t = 3.1 \)[/tex], the rate is [tex]\( 2 \cdot 3.1 = 6.2 \)[/tex].
Comparison: The average rate of change is 6.1, while the instantaneous rates at the endpoints are 6 and 6.2.
### Exercise 93: [tex]\( f(x) = \frac{-1}{x} \)[/tex], Interval: [tex]\([1, 2]\)[/tex]
1. Average Rate of Change:
- Calculate [tex]\( f(2) \)[/tex] and [tex]\( f(1) \)[/tex]:
[tex]\[
f(2) = \frac{-1}{2} = -0.5
\][/tex]
[tex]\[
f(1) = \frac{-1}{1} = -1
\][/tex]
- The average rate of change is:
[tex]\[
\frac{f(2) - f(1)}{2 - 1} = \frac{-0.5 + 1}{1} = 0.5
\][/tex]
2. Instantaneous Rates of Change:
- The derivative of [tex]\( f(x) = \frac{-1}{x} \)[/tex] is [tex]\( f'(x) = \frac{1}{x^2} \)[/tex].
- At [tex]\( x = 1 \)[/tex], the rate is [tex]\( 1 \)[/tex].
- At [tex]\( x = 2 \)[/tex], the rate is [tex]\( \frac{1}{4} \)[/tex].
Comparison: The average rate of change is 0.5, while the instantaneous rates at the endpoints are 1 and 0.25.
### Exercise 94: [tex]\( f(x) = \sin x \)[/tex], Interval: [tex]\([0, \frac{\pi}{6}]\)[/tex]
1. Average Rate of Change:
- Calculate [tex]\( f(\frac{\pi}{6}) \)[/tex] and [tex]\( f(0) \)[/tex]:
[tex]\[
f\left(\frac{\pi}{6}\right) = \sin\left(\frac{\pi}{6}\right) = 0.5
\][/tex]
[tex]\[
f(0) = \sin(0) = 0
\][/tex]
- The average rate of change is:
[tex]\[
\frac{f\left(\frac{\pi}{6}\right) - f(0)}{\frac{\pi}{6} - 0} = \frac{0.5 - 0}{\frac{\pi}{6}} = \frac{3}{\pi} \approx 0.9549
\][/tex]
2. Instantaneous Rates of Change:
- The derivative of [tex]\( f(x) = \sin x \)[/tex] is [tex]\( f'(x) = \cos x \)[/tex].
- At [tex]\( x = 0 \)[/tex], the rate is [tex]\( \cos(0) = 1 \)[/tex].
- At [tex]\( x = \frac{\pi}{6} \)[/tex], the rate is [tex]\( \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} \approx 0.866 \)[/tex].
Comparison: The average rate of change is approximately 0.9549, with instantaneous rates at the endpoints being 1 and approximately 0.866.
