Math / Algebra

QuestionGiven the graphs of $y = f(x)$ and $y = g(x)$ shown below and $h(x) = \frac{f(x)}{g(x)}$ , determine the value of $h(5)$ .

Studdy Solution

STEP 1

What is this asking? We need to find $h'(5)$ given that $h(x) = \frac{f(x)}{g(x)}$ and the graphs of $f(x)$ and $g(x)$ . Watch out! Don't forget that $h'(x)$ is the derivative of $h(x)$ , and we'll need to use the quotient rule!
Also, we're dealing with graphs, so we'll need to find the values and slopes visually.

STEP 2

1. Define the function
2. Find the derivative
3. Evaluate at $x = 5$

STEP 3

We are given that $h(x) = \frac{f(x)}{g(x)}$ .
This tells us how $h(x)$ is related to $f(x)$ and $g(x)$ .

STEP 4

To find $h'(x)$ , we need to use the quotient rule.
Remember, the quotient rule says that if $h(x) = \frac{f(x)}{g(x)}$ , then $h'(x) = \frac{g(x) \cdot f'(x) - f(x) \cdot g'(x)}{(g(x))^2}$ .

STEP 5

We want to find $h'(5)$ , so we substitute $x = 5$ into our expression for $h'(x)$ : $h'(5) = \frac{g(5) \cdot f'(5) - f(5) \cdot g'(5)}{(g(5))^2}$

STEP 6

Now, let's find the individual values.
From the graph of $f(x)$ , we see that $f(5) = -2$ .
The slope of $f(x)$ at $x=5$ is $f'(5) = \frac{-8 - (-2)}{10 - 5} = \frac{-6}{5} = -\frac{6}{5}$ .

STEP 7

From the graph of $g(x)$ , we see that $g(5) = -1$ .
The slope of $g(x)$ at $x=5$ is $g'(5) = \frac{-3 - (-1)}{6 - 5} = \frac{-2}{1} = -2$ .

STEP 8

Substitute these values into our expression for $h'(5)$ : $h'(5) = \frac{(-1) \cdot \left( -\frac{6}{5} \right) - (-2) \cdot (-2)}{(-1)^2}$ $h'(5) = \frac{\frac{6}{5} - 4}{1}$ $h'(5) = \frac{6}{5} - \frac{20}{5}$ $h'(5) = -\frac{14}{5}$

STEP 9

Therefore, $h'(5) = -\frac{14}{5}$ .

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QuestionGiven the graphs of $y = f(x)$ and $y = g(x)$ shown below and $h(x) = \frac{f(x)}{g(x)}$ , determine the value of $h(5)$ .

Studdy Solution

STEP 1

STEP 2

1. Define the function
2. Find the derivative
3. Evaluate at $x = 5$

STEP 3

We are given that $h(x) = \frac{f(x)}{g(x)}$ .
This tells us how $h(x)$ is related to $f(x)$ and $g(x)$ .

STEP 4

To find $h'(x)$ , we need to use the quotient rule.
Remember, the quotient rule says that if $h(x) = \frac{f(x)}{g(x)}$ , then $h'(x) = \frac{g(x) \cdot f'(x) - f(x) \cdot g'(x)}{(g(x))^2}$ .

STEP 5

We want to find $h'(5)$ , so we substitute $x = 5$ into our expression for $h'(x)$ : $h'(5) = \frac{g(5) \cdot f'(5) - f(5) \cdot g'(5)}{(g(5))^2}$

STEP 6

Now, let's find the individual values.
From the graph of $f(x)$ , we see that $f(5) = -2$ .
The slope of $f(x)$ at $x=5$ is $f'(5) = \frac{-8 - (-2)}{10 - 5} = \frac{-6}{5} = -\frac{6}{5}$ .

STEP 7

From the graph of $g(x)$ , we see that $g(5) = -1$ .
The slope of $g(x)$ at $x=5$ is $g'(5) = \frac{-3 - (-1)}{6 - 5} = \frac{-2}{1} = -2$ .

STEP 8

Substitute these values into our expression for $h'(5)$ : $h'(5) = \frac{(-1) \cdot \left( -\frac{6}{5} \right) - (-2) \cdot (-2)}{(-1)^2}$ $h'(5) = \frac{\frac{6}{5} - 4}{1}$ $h'(5) = \frac{6}{5} - \frac{20}{5}$ $h'(5) = -\frac{14}{5}$

STEP 9

Therefore, $h'(5) = -\frac{14}{5}$ .

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QuestionGiven the graphs of y=f(x)y = f(x)y=f(x) and y=g(x)y = g(x)y=g(x) shown below and h(x)=f(x)g(x)h(x) = \frac{f(x)}{g(x)}h(x)=g(x)f(x)​, determine the value of h(5)h(5)h(5).

Studdy Solution

STEP 1

STEP 2

1. Define the function2. Find the derivative3. Evaluate at x=5x = 5x=5

STEP 3

We are given that h(x)=f(x)g(x)h(x) = \frac{f(x)}{g(x)}h(x)=g(x)f(x)​.This tells us how h(x)h(x)h(x) is related to f(x)f(x)f(x) and g(x)g(x)g(x).

STEP 4

STEP 5

We want to find h′(5)h'(5)h′(5), so we **substitute** x=5x = 5x=5 into our expression for h′(x)h'(x)h′(x): h′(5)=g(5)⋅f′(5)−f(5)⋅g′(5)(g(5))2h'(5) = \frac{g(5) \cdot f'(5) - f(5) \cdot g'(5)}{(g(5))^2}h′(5)=(g(5))2g(5)⋅f′(5)−f(5)⋅g′(5)​

STEP 6

STEP 7

From the graph of g(x)g(x)g(x), we see that g(5)=−1g(5) = -1g(5)=−1.The slope of g(x)g(x)g(x) at x=5x=5x=5 is g′(5)=−3−(−1)6−5=−21=−2g'(5) = \frac{-3 - (-1)}{6 - 5} = \frac{-2}{1} = -2g′(5)=6−5−3−(−1)​=1−2​=−2.

STEP 8

STEP 9

Therefore, h′(5)=−145h'(5) = -\frac{14}{5}h′(5)=−514​.

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QuestionGiven the graphs of $y = f(x)$ and $y = g(x)$ shown below and $h(x) = \frac{f(x)}{g(x)}$ , determine the value of $h(5)$ .

1. Define the function
2. Find the derivative
3. Evaluate at $x = 5$

We are given that $h(x) = \frac{f(x)}{g(x)}$ .
This tells us how $h(x)$ is related to $f(x)$ and $g(x)$ .

We want to find $h'(5)$ , so we substitute $x = 5$ into our expression for $h'(x)$ : $h'(5) = \frac{g(5) \cdot f'(5) - f(5) \cdot g'(5)}{(g(5))^2}$

From the graph of $g(x)$ , we see that $g(5) = -1$ .
The slope of $g(x)$ at $x=5$ is $g'(5) = \frac{-3 - (-1)}{6 - 5} = \frac{-2}{1} = -2$ .

Therefore, $h'(5) = -\frac{14}{5}$ .