Chapter 3

Find \(d s / d t\) when \(\theta=3 \pi / 2\) if \(s=\cos \theta\) and \(d \theta / d t=5\).

Use logarithmic differentiation to find the derivative of \(y\) with respect to the given independent variable. $$y=x^{(x+1)}$$

Find \(d y / d t\) when \(x=1\) if \(y=x^{2}+7 x-5\) and \(d x / d t=1 / 3\).

Use logarithmic differentiation to find the derivative of \(y\) with respect to the given independent variable. $$y=(\sqrt{t})^{t}$$

Use logarithmic differentiation to find the derivative of \(y\) with respect to the given independent variable. $$y=t^{\sqrt{t}}$$

Use logarithmic differentiation to find the derivative of \(y\) with respect to the given independent variable. $$y=(\sin x)^{x}$$

Find the tangent to \(y=((x-1) /(x+1))^{2}\) at \(x=0\).

Use logarithmic differentiation to find the derivative of \(y\) with respect to the given independent variable. $$y=x^{\sin x}$$

Find the tangent to \(y=\sqrt{x^{2}-x+7}\) at \(x=2\).

Use logarithmic differentiation to find the derivative of \(y\) with respect to the given independent variable. $$y=x^{\ln x}$$

a. Find the tangent to the curve \(y=2 \tan (\pi x / 4)\) at \(x=1\) b. Slopes on a tangent curve What is the smallest value the slope of the curve can ever have on the interval \(-2 < x < 2 ?\) Give reasons for your answer.

Use logarithmic differentiation to find the derivative of \(y\) with respect to the given independent variable. $$y=(\ln x)^{\ln x}$$

a. Find equations for the tangents to the curves \(y=\sin 2 x\) and \(y=-\sin (x / 2)\) at the origin. Is there anything special about how the tangents are related? Give reasons for your answer. b. Can anything be said about the tangents to the curves \(y=\sin m x\) and \(y=-\sin (x / m)\) at the origin ( \(m\) a constant \(\neq 0\) )? Give reasons for your answer. c. For a given \(m,\) what are the largest values the slopes of the curves \(y=\sin m x\) and \(y=-\sin (x / m)\) can ever have? Give reasons for your answer. d. The function \(y=\sin x\) completes one period on the interval \([0,2 \pi],\) the function \(y=\sin 2 x\) completes two periods, the function \(y=\sin (x / 2)\) completes half a period, and so on. Is there any relation between the number of periods \(y=\sin m x\) completes on \([0,2 \pi]\) and the slope of the curve \(y=\sin m x\) at the origin? Give reasons for your answer.

If we write \(g(x)\) for \(f^{-1}(x),\) Equation (1) can be written as $$g^{\prime}(f(a))=\frac{1}{f^{\prime}(a)}, \quad \text { or } \quad g^{\prime}(f(a)) \cdot f^{\prime}(a)=1$$. If we then write \(x\) for \(a\), we get $$g^{\prime}(f(x)) \cdot f^{\prime}(x)=1$$. The latter equation may remind you of the Chain Rule, and indeed there is a connection. Assume that \(f\) and \(g\) are differentiable functions that are inverses of one another, so that \((g \circ f)(x)=x .\) Differentiate both sides of this equation with respect to \(x\), using the Chain Rule to express \((g \circ f)^{\prime}(x)\) as a product of derivatives of \(g\) and \(f\) What do you find? (This is not a proof of Theorem 3 because we assume here the theorem's conclusion that \(g=f^{-1}\) is differentiable.)

Suppose that a piston is moving straight up and down and that its position at time \(t\) sec is $$s=A \cos (2 \pi b t)$$ with \(A\) and \(b\) positive. The value of \(A\) is the amplitude of the motion, and \(b\) is the frequency (number of times the piston moves up and down each second). What effect does doubling the frequency have on the piston's velocity, acceleration, and jerk? (Once you find out, you will know why some machinery breaks when you run it too fast.)

$$\text { Show that } \lim _{n \rightarrow \infty}\left(1+\frac{x}{n}\right)^{n}=e^{x} \text { for any } x>0$$.

If \(f(x)=x^{n}, n \geq 1,\) show from the definition of the derivative that \(f^{\prime}(0)=0\).

The position of a particle moving along a coordinate line is \(s=\sqrt{1+4 t},\) with \(s\) in meters and \(t\) in seconds. Find the particle's velocity and acceleration at \(t=6\) sec.

Using mathematical induction, show that for \(n>1\) $$\frac{d^{n}}{d x^{n}} \ln x=(-1)^{n-1} \frac{(n-1) !}{x^{n}}$$

Suppose that the velocity of a falling body is \(v=k \sqrt{s} \mathrm{m} / \mathrm{sec}(k \text { a constant })\) at the instant the body has fallen \(s\) m from its starting point. Show that the body's acceleration is constant.

The velocity of a heavy meteorite entering Earth's atmosphere is inversely proportional to \(\sqrt{s}\) when it is \(s \mathrm{km}\) from Earth's center. Show that the meteorite's acceleration is inversely proportional to \(s^{2}\).

A particle moves along the \(x\) -axis with velocity \(d x / d t=f(x) .\) Show that the particle's acceleration is \(f(x) f^{\prime}(x)\).

You will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS: a. Plot the function \(y=f(x)\) together with its derivative over the given interval. Explain why you know that \(f\) is one-to-one over the interval. b. Solve the equation \(y=f(x)\) for \(x\) as a function of \(y,\) and name the resulting inverse function \(g\). c. Find the equation for the tangent line to \(f\) at the specified point \(\left(x_{0}, f\left(x_{0}\right)\right)\) d. Find the equation for the tangent line to \(g\) at the point \(\left(f\left(x_{0}\right), x_{0}\right)\) located symmetrically across the \(45^{\circ}\) line \(y=x\) (which is the graph of the identity function). Use Theorem 3 to find the slope of this tangent line. e. Plot the functions \(f\) and \(g\), the identity, the two tangent lines, and the line segment joining the points \(\left(x_{0}, f\left(x_{0}\right)\right)\) and \(\left(f\left(x_{0}\right), x_{0}\right)\) Discuss the symmetries you see across the main diagonal. $$y=\frac{4 x}{x^{2}+1}, \quad-1 \leq x \leq 1, \quad x_{0}=1 / 2$$

For oscillations of small amplitude (short swings), we may safely model the relationship between the period \(T\) and the length \(L\) of a simple pendulum with the equation $$T=2 \pi \sqrt{\frac{L}{g}}$$ where \(g\) is the constant acceleration of gravity at the pendulum's location. If we measure \(g\) in centimeters per second squared, we measure \(L\) in centimeters and \(T\) in seconds. If the pendulum is made of metal, its length will vary with temperature, either increasing or decreasing at a rate that is roughly proportional to L. In symbols, with \(u\) being temperature and \(k\) the proportionality constant, $$\frac{d L}{d u}=k L$$ Assuming this to be the case, show that the rate at which the period changes with respect to temperature is \(k T / 2\).

You will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS: a. Plot the function \(y=f(x)\) together with its derivative over the given interval. Explain why you know that \(f\) is one-to-one over the interval. b. Solve the equation \(y=f(x)\) for \(x\) as a function of \(y,\) and name the resulting inverse function \(g\). c. Find the equation for the tangent line to \(f\) at the specified point \(\left(x_{0}, f\left(x_{0}\right)\right)\) d. Find the equation for the tangent line to \(g\) at the point \(\left(f\left(x_{0}\right), x_{0}\right)\) located symmetrically across the \(45^{\circ}\) line \(y=x\) (which is the graph of the identity function). Use Theorem 3 to find the slope of this tangent line. e. Plot the functions \(f\) and \(g\), the identity, the two tangent lines, and the line segment joining the points \(\left(x_{0}, f\left(x_{0}\right)\right)\) and \(\left(f\left(x_{0}\right), x_{0}\right)\) Discuss the symmetries you see across the main diagonal. $$y=\frac{x^{3}}{x^{2}+1}, \quad-1 \leq x \leq 1, \quad x_{0}=1 / 2$$

Suppose that \(f(x)=x^{2}\) and \(g(x)=|x| .\) Then the composites $$(f \circ g)(x)=|x|^{2}=x^{2} \quad \text { and } \quad(g \circ f)(x)=\left|x^{2}\right|=x^{2}$$ are both differentiable at \(x=0\) even though \(g\) itself is not differentiable at \(x=0 .\) Does this contradict the Chain Rule? Explain.

You will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS: a. Plot the function \(y=f(x)\) together with its derivative over the given interval. Explain why you know that \(f\) is one-to-one over the interval. b. Solve the equation \(y=f(x)\) for \(x\) as a function of \(y,\) and name the resulting inverse function \(g\). c. Find the equation for the tangent line to \(f\) at the specified point \(\left(x_{0}, f\left(x_{0}\right)\right)\) d. Find the equation for the tangent line to \(g\) at the point \(\left(f\left(x_{0}\right), x_{0}\right)\) located symmetrically across the \(45^{\circ}\) line \(y=x\) (which is the graph of the identity function). Use Theorem 3 to find the slope of this tangent line. e. Plot the functions \(f\) and \(g\), the identity, the two tangent lines, and the line segment joining the points \(\left(x_{0}, f\left(x_{0}\right)\right)\) and \(\left(f\left(x_{0}\right), x_{0}\right)\) Discuss the symmetries you see across the main diagonal. $$y=x^{3}-3 x^{2}-1, \quad 2 \leq x \leq 5, \quad x_{0}=\frac{27}{10}$$

Graph the function \(y=2 \cos 2 x\) for \(-2 \leq x \leq 3.5 .\) Then, on the same screen, graph $$y=\frac{\sin 2(x+h)-\sin 2 x}{h}$$ for \(h=1.0,0.5,\) and \(0.2 .\) Experiment with other values of \(h\) including negative values. What do you see happening as \(h \rightarrow 0 ?\) Explain this behavior.

You will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS: a. Plot the function \(y=f(x)\) together with its derivative over the given interval. Explain why you know that \(f\) is one-to-one over the interval. b. Solve the equation \(y=f(x)\) for \(x\) as a function of \(y,\) and name the resulting inverse function \(g\). c. Find the equation for the tangent line to \(f\) at the specified point \(\left(x_{0}, f\left(x_{0}\right)\right)\) d. Find the equation for the tangent line to \(g\) at the point \(\left(f\left(x_{0}\right), x_{0}\right)\) located symmetrically across the \(45^{\circ}\) line \(y=x\) (which is the graph of the identity function). Use Theorem 3 to find the slope of this tangent line. e. Plot the functions \(f\) and \(g\), the identity, the two tangent lines, and the line segment joining the points \(\left(x_{0}, f\left(x_{0}\right)\right)\) and \(\left(f\left(x_{0}\right), x_{0}\right)\) Discuss the symmetries you see across the main diagonal. $$y=2-x-x^{3}, \quad-2 \leq x \leq 2, \quad x_{0}=\frac{3}{2}$$

Graph \(y=-2 x \sin \left(x^{2}\right)\) for \(-2 \leq\) \(x \leq 3 .\) Then, on the same screen, graph $$y=\frac{\cos \left((x+h)^{2}\right)-\cos \left(x^{2}\right)}{h}$$ for \(h=1.0,0.7,\) and \(0.3 .\) Experiment with other values of \(h\) What do you see happening as \(h \rightarrow 0 ?\) Explain this behavior.

You will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS: a. Plot the function \(y=f(x)\) together with its derivative over the given interval. Explain why you know that \(f\) is one-to-one over the interval. b. Solve the equation \(y=f(x)\) for \(x\) as a function of \(y,\) and name the resulting inverse function \(g\). c. Find the equation for the tangent line to \(f\) at the specified point \(\left(x_{0}, f\left(x_{0}\right)\right)\) d. Find the equation for the tangent line to \(g\) at the point \(\left(f\left(x_{0}\right), x_{0}\right)\) located symmetrically across the \(45^{\circ}\) line \(y=x\) (which is the graph of the identity function). Use Theorem 3 to find the slope of this tangent line. e. Plot the functions \(f\) and \(g\), the identity, the two tangent lines, and the line segment joining the points \(\left(x_{0}, f\left(x_{0}\right)\right)\) and \(\left(f\left(x_{0}\right), x_{0}\right)\) Discuss the symmetries you see across the main diagonal. $$y=e^{x}, \quad-3 \leq x \leq 5, \quad x_{0}=1$$

Using the Chain Rule, show that the Power Rule \((d / d x) x^{n}=n x^{n-1}\) holds for the functions \(x^{n}\) in Exercises 107 and 108. $$x^{1 / 4}=\sqrt{\sqrt{x}}$$

You will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS: a. Plot the function \(y=f(x)\) together with its derivative over the given interval. Explain why you know that \(f\) is one-to-one over the interval. b. Solve the equation \(y=f(x)\) for \(x\) as a function of \(y,\) and name the resulting inverse function \(g\). c. Find the equation for the tangent line to \(f\) at the specified point \(\left(x_{0}, f\left(x_{0}\right)\right)\) d. Find the equation for the tangent line to \(g\) at the point \(\left(f\left(x_{0}\right), x_{0}\right)\) located symmetrically across the \(45^{\circ}\) line \(y=x\) (which is the graph of the identity function). Use Theorem 3 to find the slope of this tangent line. e. Plot the functions \(f\) and \(g\), the identity, the two tangent lines, and the line segment joining the points \(\left(x_{0}, f\left(x_{0}\right)\right)\) and \(\left(f\left(x_{0}\right), x_{0}\right)\) Discuss the symmetries you see across the main diagonal. $$y=\sin x, \quad-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}, \quad x_{0}=1$$