Chapter 3

A man on a dock is pulling in a rope attached to a rowboat at a rate of 5 feet per second. If the man's hands are 8 feet higher than the point where the rope is attached to the boat, how fast is the angle of depression of the rope changing when there are still 17 feet of rope out?

$$ \underline{\phantom{xxx}} , \text { find the indicated derivative. } $$ $$ D_{\theta} \sqrt{\log _{10}\left(3^{\theta^{2}-\theta}\right)} $$

Suppose that curves \(C_{1}\) and \(C_{2}\) intersect at \(\left(x_{0}, y_{0}\right)\) with slopes \(m_{1}\) and \(m_{2}\), respectively, as in Figure 4 . Then (see Problem 40 of Section \(1.8\) ) the positive angle \(\theta\) from \(C_{1}\) (i.e., from the tangent line to \(C_{1}\) at \(\left.\left(x_{0}, y_{0}\right)\right)\) to \(C_{2}\) satisfies $$ \tan \theta=\frac{m_{2}-m_{1}}{1+m_{1} m_{2}} $$ Find the angles from the circle \(x^{2}+y^{2}=1\) to the circle \((x-1)^{2}+y^{2}=1\) at the two points of intersection.

Find \(D_{x} y\) using the rules of this section. $$ y=\frac{x^{2}-2 x+5}{x^{2}+2 x-3} $$

The graph of a function \(y=f(x)\) is given. Use this graph to sketch the graph of \(y=f^{\prime}(x)\).

A visitor from outer space is approaching the earth (radius \(=6376\) kilometers \()\) at 2 kilometers per second. How fast is the angle \(\theta\) subtended by the earth at her eye increasing when she is 3000 kilometers from the surface?

$$ \underline{\phantom{xxx}} , \text { find the indicated derivative. } $$ $$ D_{x}\left(10^{\left(x^{2}\right)}+\left(x^{2}\right)^{10}\right) $$

Find the angle from the line \(y=2 x\) to the curve \(x^{2}-x y+2 y^{2}=28\) at their point of intersection in the first quadrant (see Problem 44).

If \(f(0)=4, f^{\prime}(0)=-1, g(0)=-3\), and \(g^{\prime}(0)=5\), find (a) \((f \cdot g)^{\prime}(0)\) (b) \((f+g)^{\prime}(0)\) (c) \((f / g)^{\prime}(0)\)

Show that for every \(a>0\) the linear approximation \(L(x)\) to the function \(f(x)=\sqrt{x}\) at \(a\) satisfies \(f(x) \leq L(x)\) for all \(x>0\).

$$ \underline{\phantom{xxx}} , \text { find the indicated derivative. } $$ $$ D_{x}\left(\sin ^{2} x+2^{\sin x}\right) $$

A particle of mass \(m\) moves along the \(x\) -axis so that its position \(x\) and velocity \(v=d x / d t\) satisfy $$ m\left(v^{2}-v_{0}^{2}\right)=k\left(x_{0}^{2}-x^{2}\right) $$ where \(v_{0}, x_{0}\), and \(k\) are constants. Show by implicit differentiation that $$ m \frac{d v}{d t}=-k x $$ whenever \(v \neq 0\).

If \(f(3)=7, f^{\prime}(3)=2, g(3)=6\), and \(g^{\prime}(3)=-10\), find (a) \((f-g)^{\prime}(3)\) (b) \((f \cdot g)^{\prime}(3)\) (c) \((g / f)^{\prime}(3)\)

Find \(\Delta y\) for the given values of \(x_{1}\) and \(x_{2}(\) see Example 7). $$ y=3 x^{2}+2 x+1, x_{1}=0.0, x_{2}=0.1 $$

Show that for every \(a\) the linear approximation \(L(x)\) to the function \(f(x)=x^{2}\) at \(a\) satisfies \(L(x) \leq f(x)\) for all \(x\).

$$ \underline{\phantom{xxx}} , \text { find the indicated derivative. } $$ $$ D_{x}\left[x^{\pi+1}+(\pi+1)^{x}\right] $$

Express the indicated derivative in terms of the function \(F(x) .\) Assume that \(F\) is differentiable. $$ D_{x}(F(2 x)) $$

Use the Product Rule to show that \(D_{x}[f(x)]^{2}=\) \(2 \cdot f(x) \cdot D_{x} f(x)\)

Find a linear approximation to \(f(x)=(1+x)^{\alpha}\) at \(x=0\), where \(\alpha\) is any number. For various values of \(\alpha\), plot \(f(x)\) and its linear approximation \(L(x)\). For what values of \(\alpha\) does the linear approximation always overestimate \(f(x) ?\) For what values of \(\alpha\) does the linear approximation always underestimate \(f(x)\) ?

$$ \underline{\phantom{xxx}} , \text { find the indicated derivative. } $$ $$ D_{x}\left[2^{\left(e^{x}\right)}+\left(2^{e}\right)^{x}\right] $$

Find all points on the curve \(x^{2} y-x y^{2}=2\) where the tangent line is vertical, that is, where \(d x / d y=0\).

Express the indicated derivative in terms of the function \(F(x) .\) Assume that \(F\) is differentiable. $$ D_{x}\left(F\left(x^{2}+1\right)\right) $$

Develop a rule for \(D_{x}[f(x) g(x) h(x)]\)

Suppose \(f\) is differentiable. If we use the approximation \(f(x+h) \approx f(x)+f^{\prime}(x) h\) the error is \(\varepsilon(h)=f(x+h)-\) \(f(x)-f^{\prime}(x) h .\) Show that(a) \(\lim _{h \rightarrow 0} \varepsilon(h)=0\) and (b) \(\lim _{h \rightarrow 0} \frac{\varepsilon(h)}{h}=0\).

$$ \underline{\phantom{xxx}} , \text { find the indicated derivative. } $$ $$ D_{x}\left(x^{2}+1\right)^{\ln x} $$

Express the indicated derivative in terms of the function \(F(x) .\) Assume that \(F\) is differentiable. $$ D_{t}\left((F(t))^{-2}\right) $$

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

Find \(\Delta y\) for the given values of \(x_{1}\) and \(x_{2}(\) see Example 7). $$ y=\frac{3}{x+1}, x_{1}=2.34, x_{2}=2.31 $$

$$ \underline{\phantom{xxx}} , \text { find the indicated derivative. } $$ $$ D_{x}\left(\ln x^{2}\right)^{2 x+3} $$

Express the indicated derivative in terms of the function \(F(x) .\) Assume that \(F\) is differentiable. $$ \frac{d}{d z}\left(\frac{1}{(F(z))^{2}}\right) $$

Find the equation of the tangent line to \(y=1 /\left(x^{2}+4\right)\) at the point \((1,1 / 5)\).

$$ \underline{\phantom{xxx}} , \text { find the indicated derivative. } $$ $$ f^{\prime}(1) \text { if } f(x)=x^{\sin x} $$

Express the indicated derivative in terms of the function \(F(x) .\) Assume that \(F\) is differentiable. $$ \frac{d}{d z}(1+(F(2 z)))^{2} $$

Find all points on the graph of \(y=x^{3}-x^{2}\) where the tangent line is horizontal.

First find and simplify $$\frac{\Delta y}{\Delta x}=\frac{f(x+\Delta x)-f(x)}{\Delta x}$$ Then find \(d y / d x\) by taking the limit of your answer as \(\Delta x \rightarrow 0 .\) $$ y=x^{2} $$

Express the indicated derivative in terms of the function \(F(x) .\) Assume that \(F\) is differentiable. $$ \frac{d}{d y}\left(y^{2}+\frac{1}{F\left(y^{2}\right)}\right) $$

Find all points on the graph of \(y=\frac{1}{3} x^{3}+x^{2}-x\) where the tangent line has slope 1 .

First find and simplify $$\frac{\Delta y}{\Delta x}=\frac{f(x+\Delta x)-f(x)}{\Delta x}$$ Then find \(d y / d x\) by taking the limit of your answer as \(\Delta x \rightarrow 0 .\) $$ y=x^{3}-3 x^{2} $$

, find dy/dx by logarithmic differentiation. $$ y=\frac{x+11}{\sqrt{x^{3}-4}} $$

Express the indicated derivative in terms of the function \(F(x) .\) Assume that \(F\) is differentiable. $$ \frac{d}{d x} F(\cos x) $$

Find all points on the graph of \(y=100 / x^{5}\) where the tangent line is perpendicular to the line \(y=x\).

First find and simplify $$\frac{\Delta y}{\Delta x}=\frac{f(x+\Delta x)-f(x)}{\Delta x}$$ Then find \(d y / d x\) by taking the limit of your answer as \(\Delta x \rightarrow 0 .\) $$ y=\frac{1}{x+1} $$

, find dy/dx by logarithmic differentiation. $$ y=\left(x^{2}+3 x\right)(x-2)\left(x^{2}+1\right) $$

Express the indicated derivative in terms of the function \(F(x) .\) Assume that \(F\) is differentiable. $$ \frac{d}{d x} \cos F(x) $$

First find and simplify $$\frac{\Delta y}{\Delta x}=\frac{f(x+\Delta x)-f(x)}{\Delta x}$$ Then find \(d y / d x\) by taking the limit of your answer as \(\Delta x \rightarrow 0 .\) $$ y=1+\frac{1}{x} $$

, find dy/dx by logarithmic differentiation. $$ y=\frac{\sqrt{x+13}}{(x-4) \sqrt[3]{2 x+1}} $$

Express the indicated derivative in terms of the function \(F(x) .\) Assume that \(F\) is differentiable. $$ D_{x} \tan F(2 x) $$

The height \(s\) in feet of a ball above the ground at \(t\) seconds is given by \(s=-16 t^{2}+40 t+100\). (a) What is its instantaneous velocity at \(t=2\) ? (b) When is its instantaneous velocity 0 ?

First find and simplify $$\frac{\Delta y}{\Delta x}=\frac{f(x+\Delta x)-f(x)}{\Delta x}$$ Then find \(d y / d x\) by taking the limit of your answer as \(\Delta x \rightarrow 0 .\) $$ y=\frac{x-1}{x+1} $$

, find dy/dx by logarithmic differentiation. $$ y=\frac{\left(x^{2}+3\right)^{2 / 3}(3 x+2)^{2}}{\sqrt{x+1}} $$