Chapter 3

Find the equation of the tangent line to the graph of \(y=1+x \sin 3 x\) at \(\left(\frac{\pi}{3}, 1\right)\). Where does this line cross the \(x\)-axis?

Find all points on the graph of \(y=\sin ^{2} x\) where the tangent line has slope \(1 .\)

Find the equation of the tangent line to \(y=\) \(\left(x^{2}+1\right)^{3}\left(x^{4}+1\right)^{2}\) at \((1,32)\).

Find the equation of the tangent line to \(y=\left(x^{2}+1\right)^{-2}\) at \(\left(1, \frac{1}{4}\right)\).

Suppose that \(f(x+y)=f(x) f(y)\) for all \(x\) and \(y\). Show that if \(f^{\prime}(0)\) exists then \(f^{\prime}(a)\) exists and \(f^{\prime}(a)=f(a) f^{\prime}(0)\).

Where does the tangent line to \(y=(2 x+1)^{3}\) at \((0,1)\) cross the \(x\)-axis?

Let \(f(x)= \begin{cases}m x+b & \text { if } x<2 \\ x^{2} & \text { if } x \geq 2\end{cases}\) Determine \(m\) and \(b\) so that \(f\) is differentiable everywhere.

Where does the tangent line to \(y=\left(x^{2}+1\right)^{-2}\) at \(\left(1, \frac{1}{4}\right)\) cross the \(x\)-axis?

The symmetric derivative \(f_{s}(x)\) is defined by $$ f_{s}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x-h)}{2 h} $$ Show that if \(f^{\prime}(x)\) exists then \(f_{s}(x)\) exists, but that the converse is false.

A point \(P\) is moving in the plane so that its coordinates after \(t\) seconds are \((4 \cos 2 t, 7 \sin 2 t)\), measured in feet. (a) Show that \(P\) is following an elliptical path. Hint: Show that \((x / 4)^{2}+(y / 7)^{2}=1\), which is an equation of an ellipse. (b) Obtain an expression for \(L\), the distance of \(P\) from the origin at time \(t\). (c) How fast is the distance between \(P\) and the origin changing when \(t=\pi / 8\) ? You will need the fact that \(D_{u}(\sqrt{u})=\) \(1 /(2 \sqrt{u})\) (see Example 4 of Section 3.2).

Let \(f\) be differentiable and let \(f^{\prime}\left(x_{0}\right)=m\). Find \(f^{\prime}\left(-x_{0}\right)\) if (a) \(f\) is an odd function. (b) f is an even function.

A wheel centered at the origin and of radius 10 centimeters is rotating counterclockwise at a rate of 4 revolutions per second. A point \(P\) on the rim is at \((10,0)\) at \(t=0\). (a) What are the coordinates of \(P\) at time \(t\) seconds? (b) At what rate is \(P\) rising (or falling) at time \(t=1\) ?

Prove that the derivative of an odd function is an even function and that the derivative of an even function is an odd function. CAS Use a CAS to do Problems 71 and \(72 .\)

Draw the graphs of \(f(x)=x^{3}-4 x^{2}+3\) and its derivative \(f^{\prime}(x)\) on the interval \([-2,5]\) using the same axes. (a) Where on this interval is \(f^{\prime}(x)<0\) ? (b) Where on this interval is \(f(x)\) decreasing? (c) Make a conjecture. Experiment with other intervals and other functions to support this conjecture.

Draw the graphs of \(f(x)=x^{3}-4 x^{2}+3\) and its derivative \(f^{\prime}(x)\) on the interval \([-2,5]\) using the same axes. (a) Where on this interval is \(f^{\prime}(x)<0\) ? (b) Where on this interval is \(f(x)\) decreasing? (c) Make a conjecture. Experiment with other intervals and other functions to support this conjecture.

The dial of a standard clock has a 10 -centimeter radius. One end of an elastic string is attached to the rim at 12 and the other to the tip of the 10 -centimeter minute hand. At what rate is the string stretching at \(12: 15\) (assuming that the clock is not slowed down by this stretching)?

Let \(x_{0}\) be the smallest positive value of \(x\) at which the curves \(y=\sin x\) and \(y=\sin 2 x\) intersect. Find \(x_{0}\) and also the acute angle at which the two curves intersect at \(x_{0}\) (see Problem 40 of Section 1.8).

Show that \(D_{x}|x|=|x| / x, x \neq 0\). Hint: Write \(|x|=\sqrt{x^{2}}\) and use the Chain Rule with \(u=x^{2}\).

Let \(f(0)=1\) and \(f^{\prime}(0)=2\). Find the derivative of \(f(f(x)-1)\) at \(x=0 .\)

Let \(f(0)=0\) and \(f^{\prime}(0)=2\). Find the derivative of \(f(f(f(f(x))))\) at \(x=0 .\)

Suppose that \(f\) is a differentiable function. (a) Find \(\frac{d}{d x} f(f(x))\). (b) Find \(\frac{d}{d x} f(f(f(x)))\). (c) Let \(f^{[n]}\) denote the function defined as follows: \(f^{[1]}=f\) and \(f^{[n]}=f \circ f^{[n-1]}\) for \(n \geq 2\). Thus \(f^{[2]}=f \circ f, f^{[3]}=\) \(f \circ f \circ f\), etc. Based on your results from parts (a) and (b), make a conjecture regarding \(\frac{d}{d x} f^{[n]}\). Prove your conjecture.

Give a second proof of the Quotient Rule. Write $$ D_{x}\left(\frac{f(x)}{g(x)}\right)=D_{x}\left(f(x) \frac{1}{g(x)}\right) $$ and use the Product Rule and the Chain Rule.

Suppose that \(f\) is differentiable and that there are real numbers \(x_{1}\) and \(x_{2}\) such that \(f\left(x_{1}\right)=x_{2}\) and \(f\left(x_{2}\right)=x_{1}\). Let \(g(x)=f(f(f(f(x))))\). Show that \(g^{\prime}\left(x_{1}\right)=g^{\prime}\left(x_{2}\right)\).