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Chapter 10: Problem 121

Here is the graph of the derivative \(f^{\prime}\) of a function \(f\). Give a rough sketch of the graph of \(f\), given that \(f(0)=0\).

Short Answer

Expert verified
To sketch the graph of \(f(x)\), first identify the critical points on the graph of \(f'(x)\). Then, determine whether the function is increasing or decreasing by analyzing the sign of the derivative. Identify local maxima and minima by noting where the derivative changes from positive to negative or vice versa. Locate inflection points by observing where the function changes concavity. Finally, taking into account the initial condition \(f(0)=0\), sketch the graph of \(f(x)\) by connecting the critical points, local maxima and minima, and inflection points.

Step by step solution

01

Identify the critical points

The critical points of a function are the points where its derivative is either zero or undefined. These points often correspond to local minima, maxima, or inflection points in the original function. Locate these points on the graph of the derivative, and note their x-coordinates.
02

Determine whether the function is increasing or decreasing

Analyze the graph of the derivative to determine whether the original function is increasing or decreasing. If the graph of the derivative is positive, the original function is increasing; if the graph of the derivative is negative, the original function is decreasing. Use this information to draw arrows indicating the increase or decrease of the function between critical points.
03

Identify local maxima and minima

At each critical point, we can use the sign of the derivative before and after the point to determine whether the original function has a local maximum or minimum at that location. If the derivative switches from positive to negative at a critical point, this indicates a local maximum. If the derivative switches from negative to positive at a critical point, this indicates a local minimum.
04

Identify inflection points

Analyze the behavior of the original function based on the information gleaned from steps 2 and 3. If the function changes concavity at a specific point, we call that point an inflection point 鈥� that means the second derivative of the function changes its sign there. Inflection points often occur when the graph of the first derivative changes slope but does not cross the x-axis.
05

Draw the graph of the original function

With the information collected in steps 1 through 4, we can sketch the graph of the original function, keeping in mind the initial condition \(f(0)=0\). Plot the critical points, the increase or decrease of the function, the local maxima and minima, and the inflection points. Connect the plotted points and features to create an approximate sketch of the graph of the function \(f(x)\).

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Most popular questions from this chapter

Compute the indicated derivative. $$ U(t)=-1.3 t^{2}-4.5 t ; U^{\prime}(1) $$

(a) use any method to estimate the slope of the tangent to the graph of the given function at the point with the given \(x\) -coordinate and \((\boldsymbol{b})\) find an equation of the tangent line in part (a). In each case, sketch the curve together with the appropriate tangent line. HINT [See Example \(2(\mathrm{~b}) .]\) $$ f(x)=\frac{1}{x^{2}} ; x=1 $$

Crude Oil Prices The price per barrel of crude oil in constant 2008 dollars can be approximated by $$ P(t)=0.45 t^{2}-12 t+105 \text { dollars } \quad(0 \leq t \leq 28) $$ where \(t\) is time in years since the start of \(1980 .^{40}\) a. What, in constant 2008 dollars, was the average rate of change of the price of oil from the start of 1981 ( \(t=1\) ) to the start of \(2006(t=26)\) ? HINT [See Example 3.] b. Your answer to part (a) is quite small. Can you conclude that the price of oil hardly changed at all over the 25 -year period 1981 to 2006 ? Explain.

Calculate the average rate of change of the given function fover the intervals [a, \(a+h]\) where \(h=1,0.1\), 0.01,0.001, and 0.0001. (Technology is recommended for the cases \(h=0.01,0.001\), and \(0.0001)\) HINT [See Example 4.] \(f(x)=2 x^{2} ; a=0\)

Sketch the graph of a function whose average rate of change over \([0,3]\) is negative but whose average rate of change over \([1,3]\) is positive.
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