Chapter 3

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?

In Problems 45-50, find \(\Delta y\) for the given values of \(x_{1}\) and \(x_{2}\) (see Example 7). 45\. \(y=3 x+2, x_{1}=1, x_{2}=1.5\)

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) \((f / g)^{\prime}(0)\)

. 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 \(v \neq 0\).

Find the indicated derivative. \(D_{x}\left(\sin ^{2} x+2^{\sin x}\right)\)

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) \((g / f)^{\prime}(3)\)

The curve \(x^{2}-x y+y^{2}=16\) is an ellipse centered at the origin and with the line \(y=x\) as its major axis. Find the equations of the tangent lines at the two points where the ellipse intersects the \(x\)-axis.

Find the indicated derivative. \(D_{x}\left[x^{\pi+1}+(\pi+1)^{x}\right]\)

\(y=\frac{1}{x}, x_{1}=1.0, x_{2}=1.2\)

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

In Problems 47-58, express the indicated derivative in terms of the function \(F(x)\). Assume that \(F\) is differentiable. $$ D_{x}(F(2 x)) $$

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

Find the indicated derivative. \(D_{x}\left[2^{\left(e^{x}\right)}+\left(2^{e}\right)^{x}\right]\)

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

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

In Problems 47-58, express the indicated derivative in terms of the function \(F(x)\). Assume that \(F\) is differentiable. $$ D_{x}\left(F\left(x^{2}+1\right)\right) $$

Find the indicated derivative. \(D_{x}\left(x^{2}+1\right)^{\ln x}\)

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

In Problems 47-58, 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 indicated derivative. \(D_{x}\left(\ln x^{2}\right)^{2 x+3}\)

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

In Problems 47-58, 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 indicated derivative. \(f^{\prime}(1)\) if \(f(x)=x^{\sin x}\)

In Problems 51-56, 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 \(\Delta x \rightarrow 0\). 51\. \(y=x^{2}\)

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

In Problems 47-58, 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 the indicated derivative. \(D_{x} x^{\left(2^{x}\right)}\)

\(y=x^{3}-3 x^{2}\)

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

In Problems 47-58, 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 \(d y / d x\) by logarithmic differentiation \(y=\frac{x+11}{\sqrt{x^{3}-4}}\)

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

In Problems 47-58, express the indicated derivative in terms of the function \(F(x)\). Assume that \(F\) is differentiable. $$ \frac{d}{d x} F(\cos x) $$

Find \(d y / d x\) by logarithmic differentiation \(y=\left(x^{2}+3 x\right)(x-2)\left(x^{2}+1\right)\)

In Problems 47-58, express the indicated derivative in terms of the function \(F(x)\). Assume that \(F\) is differentiable. $$ \frac{d}{d x} \cos F(x) $$

Find \(d y / d x\) by logarithmic differentiation \(y=\frac{\sqrt{x+13}}{(x-4) \sqrt[3]{2 x+1}}\)

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 ?

In Problems 47-58, express the indicated derivative in terms of the function \(F(x)\). Assume that \(F\) is differentiable. $$ D_{x} \tan F(2 x) $$

Find \(d y / d x\) by logarithmic differentiation \(y=\frac{\left(x^{2}+3\right)^{2 / 3}(3 x+2)^{2}}{\sqrt{x+1}}\)

A ball rolls down a long inclined plane so that its distance \(s\) from its starting point after \(t\) seconds is \(s=4.5 t^{2}+2 t\) feet. When will its instantaneous velocity be 30 feet per second?

Find and simplify \(f^{\prime}(1)\) if $$ f(x)=\ln \left(\frac{a x-b}{a x+b}\right)^{c}, \text { where } c=\frac{a^{2}-b^{2}}{2 a b} . $$

57\. There are two tangent lines to the curve \(y=4 x-x^{2}\) that go through \((2,5)\). Find the equations of both of them.

In Problems 47-58, express the indicated derivative in terms of the function \(F(x)\). Assume that \(F\) is differentiable. $$ D_{x}\left(F(x) \sin ^{2} F(x)\right) $$

Convince yourself that \(f(x)=\left(x^{x}\right)^{x}\) and \(g(x)=x^{\left(x^{x}\right)}\) are not the same function. Then find \(f^{\prime}(x)\) and \(g^{\prime}(x)\). Note: When mathematicians write \(x^{x^{x}}\), they mean \(x^{\left(x^{x}\right)}\).

In Problems 47-58, express the indicated derivative in terms of the function \(F(x)\). Assume that \(F\) is differentiable. $$ D_{x} \sec ^{3} F(x) $$

Given that \(f(0)=1\) and \(f^{\prime}(0)=2\), find \(g^{\prime}(0)\) where \(g(x)=\cos f(x)\)

Let \(P(a, b)\) be a point on the first quadrant portion of the curve \(y=1 / x\) and let the tangent line at \(P\) intersect the \(x\)-axis at \(A\). Show that triangle \(A O P\) is isosceles and determine its area.

Given that \(F(0)=2\) and \(F^{\prime}(0)=-1\), find \(G^{\prime}(0)\) where \(G(x)=\frac{x}{1+\sec F(2 x)}\)

The radius of a spherical watermelon is growing at a constant rate of 2 centimeters per week. The thickness of the rind is always one-tenth of the radius. How fast is the volume of the rind growing at the end of the fifth week? Assume that the radius is initially 0 .

Given that \(f(1)=2, f^{\prime}(1)=-1, g(1)=0\) and \(g^{\prime}(1)=1\), find \(F^{\prime}(1)\) where \(F(x)=f(x) \cos g(x)\).