Question

48-52. sketching derivatives reproduce the graph of f and then plot a graph of f on the same axes. 48. y = f(x) 49. y = f(x) 50. y = f(x)

Explanation:

Step1: Recall derivative - slope relationship

The derivative $f'(x)$ at a point is the slope of the tangent line to the curve $y = f(x)$ at that point.

Step2: Analyze positive - slope regions

Where the function $y = f(x)$ is increasing, $f'(x)>0$. For example, if the curve of $y = f(x)$ is going up as $x$ increases, the slope of the tangent line is positive, so the graph of $y = f'(x)$ will be above the $x$ - axis in that interval.

Step3: Analyze negative - slope regions

Where the function $y = f(x)$ is decreasing, $f'(x)<0$. That is, when the curve of $y = f(x)$ is going down as $x$ increases, the slope of the tangent line is negative, and the graph of $y = f'(x)$ will be below the $x$ - axis in that interval.

Step4: Analyze zero - slope points

At the local maxima and minima of $y = f(x)$, the slope of the tangent line is zero, so $f'(x)=0$. These points will be the $x$ - intercepts of the graph of $y = f'(x)$.

For problem 48:
The function $y = f(x)$ is increasing on the left - hand side of its maximum and decreasing on the right - hand side. The graph of $y = f'(x)$ will be positive on the left of the $x$ - value of the maximum, zero at the maximum, and negative on the right of the maximum.

For problem 49:
The function $y = f(x)$ is increasing throughout. So the graph of $y = f'(x)$ will be positive throughout. It may start with a smaller positive value (where the function is increasing slowly) and then increase (where the function is increasing more rapidly).

For problem 50:
The function $y = f(x)$ is increasing. The graph of $y = f'(x)$ is positive. It may start with a larger positive value (where the function is increasing rapidly initially) and then decrease towards a smaller positive value (as the function's rate of increase slows down).

Since this is a graph - sketching problem and not a numerical calculation, we cannot give a single numerical answer. The general approach to sketch $f'(x)$ for each of the given $f(x)$ graphs is as described above.

Answer:

Step1: Recall derivative - slope relationship

The derivative $f'(x)$ at a point is the slope of the tangent line to the curve $y = f(x)$ at that point.

Step2: Analyze positive - slope regions

Where the function $y = f(x)$ is increasing, $f'(x)>0$. For example, if the curve of $y = f(x)$ is going up as $x$ increases, the slope of the tangent line is positive, so the graph of $y = f'(x)$ will be above the $x$ - axis in that interval.

Step3: Analyze negative - slope regions

Where the function $y = f(x)$ is decreasing, $f'(x)<0$. That is, when the curve of $y = f(x)$ is going down as $x$ increases, the slope of the tangent line is negative, and the graph of $y = f'(x)$ will be below the $x$ - axis in that interval.

Step4: Analyze zero - slope points

At the local maxima and minima of $y = f(x)$, the slope of the tangent line is zero, so $f'(x)=0$. These points will be the $x$ - intercepts of the graph of $y = f'(x)$.

For problem 48:
The function $y = f(x)$ is increasing on the left - hand side of its maximum and decreasing on the right - hand side. The graph of $y = f'(x)$ will be positive on the left of the $x$ - value of the maximum, zero at the maximum, and negative on the right of the maximum.

For problem 49:
The function $y = f(x)$ is increasing throughout. So the graph of $y = f'(x)$ will be positive throughout. It may start with a smaller positive value (where the function is increasing slowly) and then increase (where the function is increasing more rapidly).

For problem 50:
The function $y = f(x)$ is increasing. The graph of $y = f'(x)$ is positive. It may start with a larger positive value (where the function is increasing rapidly initially) and then decrease towards a smaller positive value (as the function's rate of increase slows down).

Since this is a graph - sketching problem and not a numerical calculation, we cannot give a single numerical answer. The general approach to sketch $f'(x)$ for each of the given $f(x)$ graphs is as described above.