91Ó°ÊÓ

If f is a differentiable function, then the values $x=c$at which the sign of the derivative $f\text{'}\left(x\right)$ changes are the locations of the local extrema of f. Use this information to find the local extrema of the functionrole="math" localid="1649873816717" $f\left(x\right)=\sin x$. Illustrate your answer on a graph of$y=\sin x$.

Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.

(a) The graph of a function whose associated slope function $f^{\text{'}}$is positive on (−∞, 2) and negative on (2,∞).

(b) The graph of a function with the following three properties: The average rate of change of f on [0, 2] is 3, the average rate of change of f on [0, 1] is −1, and the average rate of change of f on [−2, 2] is 0.

(c) The graph of a function f with the following three properties: The instantaneous rate of change of f at x = 2 is zero, the average rate of change of f on [1, 2] is 2, and the average rate of change of f on [2, 4] is 1.

Find the derivatives of each of the functions in excercises

Fill in the blank:

$\frac{d}{dx}\left(b^x\right)=---$ ($b$ is constant)

Find the derivatives of the functions:$f\left(x\right)=e^{3x}\ln (x^2+1)$

Find the derivatives of the function:$f\left(x\right)=x^{3}secx.$

Find the derivatives of each of the functions in Exercises 17–50. In some cases it may be convenient to do some preliminary algebra.

$f\left(x\right)={\tan }^2\left(3x+1\right).$

The derivative of a smooth function fat a point $x=c$can also be approximated with a symmetric difference quotient:

$f\text{'}\left(c\right)≈\frac{f(c+h)-f(c-h)}{2h}$

(a) Use a graph to illustrate what the symmetric difference measures. Why would it be reasonable to use the two-sided symmetric difference to approximate f'(c)?(Hint: Your answer should involve a certain kind of secant line and a discussion of what happens as h gets close to $0$.)

(b) Use a sequence of symmetric difference approximations to estimate the derivative of $f\left(x\right)=x^{2}atx=3$Illustrate your answer with a sequence of graphs.

Express the following limit statement as delta–epsilon statement:

$f\text{'}\left(3\right)=\underset{x→3}{lim} \frac{x^2−9}{x−3}$

Match the two graphs shown here to the equations $x{y}^2+x=1$and$x{y}^2+x+1=0$Explain your choices.