91Ó°ÊÓ

Sketching a Graph Sketch a graph of a function whose derivative is zero at exactly two points. Explain how you found the answer.

Tangent Lines and Normal Lines In Exercises 57 and 58 , find equations for the tangent line and normal line to the circle at each given point. (The normal line at a point is perpendicular to the tangent line at the point.) $$x^{2}+y^{2}=25$$ $$(4,3),(-3,4)$$

Tangent Lines and Normal Lines In Exercises 57 and 58 , find equations for the tangent line and normal line to the circle at each given point. (The normal line at a point is perpendicular to the tangent line at the point.) Use a graphing utility to graph the circle, the tangent lines, and the normal lines. $$x^{2}+y^{2}=36$$ $$(6,0),(5, \sqrt{11})$$

Circles Two circles of radius 4 are tangent to the graph of \(y^{2}=4 x\) at the point \((1,2) .\) Find equations of these two circles.

Vertical and Horizontal Tangent Lines In Exercises 61 and \(62,\) find the points at which the graph of the equation has a vertical or horizontal tangent line. \(25 x^{2}+16 y^{2}+200 x-160 y+400=0\)

Proof Prove (Theorem 2.3) that $$\frac{d}{d x}\left[x^{n}\right]=n x^{n-1}$$ for the case in which \(n\) is a rational number. (Hint: Write \(y=x^{p / q}\) in the form \(y q=x^{p}\) and differentiate implicitly. Assume that \(p\) and \(q\) are integers, where \(q>0 . )\)

Slope Find all points on the circle \(x^{2}+y^{2}=100\) where the slope is \(\frac{3}{4} .\)

Normal Lines \((\) a) Find an equation of the normal line to the ellipse \(\frac{x^{2}}{32}+\frac{y^{2}}{8}=1\) at the point \((4,2)\) . (b) Use a graphing utility to graph the ellipse and the normal line. (c) At what other point does the normal line intersect the ellipse?

Finding an Equation of a Tangent Line In Exercises 81 and 82 , find an equation of the tangent line to the graph of the function \(f\) through the point \(\left(x_{0}, y_{0}\right)\) not on the graph. To find the point of tangency \((x, y)\) on the graph of \(f\) , solve the equation \(f^{\prime}(x)=\frac{y_{0}-y}{x_{0}-x}\) $$\begin{array}{c}{\text f(x)=\sqrt{x}} \\ {\left(x_{0}, y_{0}\right)=(-4,0)}\end{array}$$

Horizontal Tangent Line Determine the point(s) in the interval \((0,2 \pi)\) at which the graph of \(f(x)=2 \cos x+\sin 2 x\) has a horizontal tangent.