Questions: a. Use the applet above to determine the value of the difference quotient (the average rate of change) near the point (2, f(2)) when: i. the length of h is 5. ii. the length of h is 3 iii. the length of h is 1.6 iv. the length of h is 0.5,6 v. the length of h is 0.15. b. Determine if the following statement is true or false. True ✓ infinity As the value of h gets smaller and smaller, the value of the difference quotient becomes a better and better approximation function f near the point (2, f(2)).

a. Use the applet above to determine the value of the difference quotient (the average rate of change) near the point (2, f(2)) when:
i. the length of h is 5. 
ii. the length of h is 3 
iii. the length of h is 1.6 
iv. the length of h is 0.5,6 
v. the length of h is 0.15. 
b. Determine if the following statement is true or false.
True
✓ infinity
As the value of h gets smaller and smaller, the value of the difference quotient becomes a better and better approximation function f near the point (2, f(2)).
Transcript text: a. Use the applet above to determine the value of the difference quotient (the average rate of change) near the point $(2, f(2))$ when: i. the length of $h$ is 5 . ii. the length of $h$ is 3 iii. the length of $h$ is 1.6 iv. the length of $h$ is $0.5,6$ v. the length of $h$ is 0.15 . b. Determine if the following statement is true or false. True $\checkmark \infty$ sthe value of $h$ gets smaller and smaller, the value of the difference quotient becomes a better and better approximation function $f$ near the point $(2, f(2))$.
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Solution

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Solution Steps

Step 1: Identify the given point and the function

The given point is \( (2, f(2)) \). From the graph, we can see that \( f(2) = 10 \).

Step 2: Determine the value of the difference quotient for different lengths of \( h \)

The difference quotient is given by: \[ \frac{f(2+h) - f(2)}{h} \]

Step 3: Calculate the difference quotient for \( h = 5 \)

For \( h = 5 \): \[ f(2+5) = f(7) = 50 \] \[ \frac{f(7) - f(2)}{5} = \frac{50 - 10}{5} = \frac{40}{5} = 8 \]

Final Answer

The value of the difference quotient for \( h = 5 \) is 8.

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