91Ó°ÊÓ

Writing In Exercises \(33-36\) , explain why the Mean Value Theorem does not apply to the function \(f\) on the interval \([0,6]\) . \(f(x)=|x-3|\)

Using the Second Derivative Test In Exercises \(33-44\) , find all relative extrema of the function. Use the Second Derivative Test where applicable. \(f(x)=-x^{4}+2 x^{3}+8 x\)

Using the Second Derivative Test In Exercises \(33-44\) , find all relative extrema of the function. Use the Second Derivative Test where applicable. \(f(x)=x^{2 / 3}-3\)

Tangent Lines The graph of \(f(x)=-\sin x\) has infinitely many tangent lines that pass through the origin. Use Newton's Method to approximate to three decimal places the slope of the tangent line having the greatest slope.

Surveying A surveyor standing 50 feet from the base of a large tree measures the angle of elevation to the top of the tree as \(71.5^{\circ} .\) How accurately must the angle be measured if the percent error in estimating the height of the tree is to be less than 6\(\% ?\)

Finding Extrema Using Technology In Exercises 49 and 50 , (a) use a computer algebra system to graph the function and approximate any absolute extrema on the given interval. (b) Use the utility to find any critical numbers, and use them to find any absolute extrema not located at the endpoints. Compare the results with those in part (a). \(f(x)=\frac{4}{3} x \sqrt{3-x}, \quad[0,3]\)

Using the Mean Value Theorem In Exercises \(49-52\) , use a graphing utility to (a) graph the function \(f\) on the given interval, (b) find and graph the secant line through points on the graph of \(f\) at the endpoints of the given interval, and v(c) find and graph any tangent lines to the graph of \(f\) that are parallel to the secant line. \(f(x)=x-2 \sin x, \quad[-\pi, \pi]\)

Using the Mean Value Theorem In Exercises \(49-52\) , use a graphing utility to (a) graph the function \(f\) on the given interval, (b) find and graph the secant line through points on the graph of \(f\) at the endpoints of the given interval, and v(c) find and graph any tangent lines to the graph of \(f\) that are parallel to the secant line. \(f(x)=x^{4}-2 x^{3}+x^{2},[0,6]\)

Limits Explain the differences between limits at infinity and infinite limits.

Vertical Motion The height of an object \(t\) seconds after it is dropped from a height of 300 meters is \(s(t)=-4.9 t^{2}+300\) (a) Find the average velocity of the object during the first 3 seconds. (b) Use the Mean Value Theorem to verify that at some time during the first 3 seconds of fall, the instantaneous velocity equals the average velocity. Find that time.