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\begin{tabular}{|c|c|c|c|c|c|} \hline

x

& -2 & 3 & 5 & 12 & 15 \\ \hline

f(x)

...

May 10, 2024

Solution by Steps

step 1

To find the average rate of change of

f'

over the interval

3 \leq x \leq 12

, we use the values of

f'(x)

x=3

and

x=12

step 2

The average rate of change is given by the formula

\frac{f'(x_2) - f'(x_1)}{x_2 - x_1}

step 3

Plugging in the values from the table, we get

\frac{f'(12) - f'(3)}{12 - 3} = \frac{9 - 5}{12 - 3}

step 4

Simplifying, we find the average rate of change to be

\frac{4}{9}

[a] Answer

The average rate of change of

f'

over the interval

3 \leq x \leq 12

\frac{4}{9}

Key Concept

Average Rate of Change

Explanation

The average rate of change of a function over an interval is the change in the function values divided by the change in the input values over that interval.

step 1

The function

m(x) = f(f(x))

is incorrect as per the context given. The correct function is

m(x) = f(x) - f'(x)

step 2

By the Intermediate Value Theorem, if a continuous function changes signs over an interval, it must cross zero at least once within that interval

step 3

Since

f(x)

and

f'(x)

are continuous and differentiable,

m(x)

is also continuous

step 4

We need to check the signs of

m(x)

at the endpoints of the interval

[3, 12]

step 5

Using the table,

m(3) = f(3) - f'(3) = -2 - 5 = -7

and

m(12) = f(12) - f'(12) = 15 - 9 = 6

step 6

Since

m(3)

is negative and

m(12)

is positive, there must be at least one value of

x

[3, 12]

where

m(x) = 0

[b] Answer

There must be at least one value of

x

[3, 12]

such that

m(x) = 0

Key Concept

Intermediate Value Theorem

Explanation

The Intermediate Value Theorem states that if a function is continuous on a closed interval and takes on different signs at the endpoints, then it must have a zero within that interval.

step 1

To approximate the integral using a right Riemann sum, we multiply the height of each subinterval by the width of that subinterval

step 2

The subintervals are

[3,5]

[5,12]

, and

[12,15]

. We use the values of

f''(x)

at the right endpoints for the heights

step 3

The approximation is

\sum (f''(x_i) \cdot \Delta x_i)

where

x_i

are the right endpoints and

\Delta x_i

are the widths of the subintervals

step 4

Calculating, we get

(-1 \cdot 2) + (2 \cdot 7) + (-6 \cdot 3)

step 5

Simplifying, the approximation is

-2 + 14 - 18 = -6

[c] Answer

The approximate value of the definite integral from 3 to 15 of

f''(x) dx

using a right Riemann sum is

-6

Key Concept

Right Riemann Sum

Explanation

A right Riemann sum approximates the value of a definite integral by summing the products of the function values at the right endpoints of subintervals and the widths of those subintervals.

step 1

To find

g(5)

, we need to evaluate the integral from

-2

3 \cdot 5 = 15

f'(t) dt

step 2

We can use the Fundamental Theorem of Calculus and the values of

f(x)

from the table since

f'(t) dt

is the derivative of

f(t)

step 3

The integral from

-2

15

f'(t) dt

f(15) - f(-2)

step 4

Using the table, we get

f(15) - f(-2) = 36 - (-10) = 46

step 5

Therefore,

g(5) = 7 + 46 = 53

step 6

To find

g'(5)

, we differentiate

g(x)

with respect to

x

step 7

By the Chain Rule,

g'(x) = \frac{d}{dx} \left(7 + \int_{-2}^{3x} f'(t) dt\right) = 3f'(3x)

step 8

Plugging in

x=5

, we get

g'(5) = 3f'(15)

step 9

Using the table,

f'(15) = 2

, so

g'(5) = 3 \cdot 2 = 6

[d] Answer

g(5) = 53

and

g'(5) = 6

Key Concept

Fundamental Theorem of Calculus and Chain Rule

Explanation

The Fundamental Theorem of Calculus relates the integral of a function to its antiderivative, and the Chain Rule is used to differentiate composite functions.