Chapter 4

Radioactive Decay Suppose \(W(t)\) denotes the amount of a radioactive material left after time \(t\) (measured in days). Assume that the half-life of the material is 3 days. Find the differential equation for the radioactive decay function \(W(t)\).

Differentiate with respect to the independent variable. $$ g(s)=\frac{s^{1 / 7}-s^{2 / 7}}{s^{3 / 7}+s^{4 / 7}} $$

Use logarithmic differentiation to find the first derivative of the given functions. $$ f(x)=x^{2 \ln x} $$

In Problems , find the coordinates of all of the points of the graph of \(y=f(x)\) that have horizontal tangents.$$ f(x)=-4 x^{4}+x^{3} $$

Graph each function and, on the basis of the graph, guess where the function is not differentiable. (Assume the largest possible domain.) $$ f(x)=\left\\{\begin{array}{cl} x^{2}+1 & \text { for } x \leq 0 \\ e^{-x} & \text { for } x>0 \end{array}\right. $$

Suppose that water is stored in a cylindrical tank of radius 5 \(\mathrm{m}\). If the height of the water in the tank is \(h\), then the volume of the water is \(V=\pi r^{2} h=\left(25 \mathrm{~m}^{2}\right) \pi h=25 \pi h \mathrm{~m}^{2} .\) If we drain the water at a rate of 250 liters per minute, what is the rate at which the water level inside the tank drops? (Note that 1 cubic meter contains 1000 liters.)

Find the derivative of $$ f(x)=\sin ^{2}\left(x^{2}-1\right) $$

Radioactive Decay Suppose \(W(t)\) denotes the amount of a radioactive material left after time \(t\) (measured in days). Assume that the half-life of the material is 5 days. Find the differential equation for the radioactive decay function \(W(t) .\)

Differentiate with respect to the independent variable. $$ f(x)=(1-2 x)\left(\sqrt{2 x}+\frac{2}{\sqrt{x}}\right) $$

Use logarithmic differentiation to find the first derivative of the given functions. $$ f(x)=x^{1 / x} $$

In Problems , find the coordinates of all of the points of the graph of \(y=f(x)\) that have horizontal tangents. $$ f(x)=\frac{1}{2} x^{4}-\frac{7}{3} x^{3}-2 x^{2} $$

Suppose the function \(f(x)\) is piecewise defined; that is, \(f(x)=f_{1}(x)\) for \(x \leq a\) and \(f(x)=f_{2}(x)\) for \(x>a\). Assume that \(f_{1}(x)\) is continuous and differentiable for \(xa\). Sketch graphs of \(f(x)\) for the following three cases: (a) \(f(x)\) is continuous and differentiable at \(x=a\). (b) \(f(x)\) is continuous, but not differentiable, at \(x=a\). (c) \(f(x)\) is neither continuous nor differentiable at \(x=a\).

Suppose that we pump water into an inverted right circular conical tank at the rate of 5 cubic feet per minute (i.e., the tank stands with its point facing downward). The tank has a height of 6 ft and the radius on top is \(3 \mathrm{ft}\). What is the rate at which the water level is rising when the water is \(2 \mathrm{ft}\) deep? (Note that the volume of a right circular cone of radius \(r\) and height \(h\) is \(V=\frac{1}{3} \pi r^{2} h .\) )

Find the derivative of $$ f(x)=\cos ^{2}\left(2 x^{2}+3\right) $$

Radioactive Decay Suppose \(W(t)\) denotes the amount of a radioactive material left after time \(t\). Assume that \(W(0)=15\) and that $$ \frac{d W}{d t}=-2 W(t) $$ (a) How much material is left at time \(t=2 ?\) (b) What is the half-life of this material?

Differentiate with respect to the independent variable. $$ f(x)=\left(x^{3}-3 x^{2}+2\right)\left(\sqrt{x}+\frac{1}{\sqrt{x}}-1\right) $$

Use logarithmic differentiation to find the first derivative of the given functions. $$ f(x)=x^{3 / x} $$

In Problems , find the coordinates of all of the points of the graph of \(y=f(x)\) that have horizontal tangents. $$ f(x)=3 x^{5}-\frac{3}{2} x^{4} $$

Two people start biking from the same point. One bikes east at \(15 \mathrm{mph}\), the other south at \(18 \mathrm{mph}\). What is the rate at which the distance between the two people is changing after 20 minutes and after 40 minutes?

Find the derivative of $$ f(x)=\tan ^{3}\left(3 x^{3}-3\right) $$

Radioactive Decay Suppose \(W(t)\) denotes the amount of a radioactive material left after time \(t\). Assume that \(W(0)=6\) and that $$ \frac{d W}{d t}=-3 W(t) $$ (a) How much material is left at time \(t=4 ?\) (b) What is the half-life of the material?

In Problems \(71-74\), find the tangent line, in slope-intercept form, of \(y=f(x)\) at the specified point. $$ f(x)=\frac{x^{2}+3}{x^{3}+5}, \text { at } x=-2 $$

Use logarithmic differentiation to find the first derivative of the given functions. $$ y=x^{x^{x}} $$

Allometric equations describe the scaling relationship between two measurements, such as skull length versus body length. In vertebrates, we typically find that [skull length] \(\propto\) [body length] \(^{a}\) for \(0

Find the derivative of $$ f(x)=\sec ^{2}\left(2 x^{2}-2\right) $$

Radioactive Decay Suppose \(W(t)\) denotes the amount of a radioactive material left after time \(t .\) Assume that \(W(0)=10\) and \(W(1)=8\) (a) Find the differential equation that describes this situation. (b) How much material is left at time \(t=5 ?\) (c) What is the half-life of the material?

Find the tangent line, in slope-intercept form, of \(y=f(x)\) at the specified point. $$ f(x)=\frac{3}{x}-\frac{4}{\sqrt{x}}+\frac{2}{x^{2}}, \text { at } x=1 $$

Use logarithmic differentiation to find the first derivative of the given functions. $$ y=\left(x^{x}\right)^{x} $$

Find the first and the second derivatives of each function. \(f(x)=x^{3}-3 x^{2}+1\)

Radioactive Decay Suppose \(W(t)\) denotes the amount of a radioactive material left after time \(t .\) Assume that \(W(0)=5\) and \(W(1)=2\) (a) Find the differential equation that describes this situation. (b) How much material is left at time \(t=3\) ? (c) What is the half-life of the material?

Find the tangent line, in slope-intercept form, of \(y=f(x)\) at the specified point. $$ f(x)=\frac{2 x-5}{x^{3}}, \text { at } x=2 \quad \text { 74. } f(x)=\sqrt{x}\left(x^{3}-1\right), \text { at } x=1 $$

Use logarithmic differentiation to find the first derivative of the given functions. $$ y=x^{\cos x} $$

Find the first and the second derivatives of each function. \(f(x)=\left(2 x^{2}+4\right)^{3}\)

Find the tangent line, in slope-intercept form, of \(y=f(x)\) at the specified point. $$ f(x)=\sqrt{x}\left(x^{3}-1\right), \text { at } x=1 $$

Use logarithmic differentiation to find the first derivative of the given functions. $$ y=(\cos x)^{x} $$

Find the first and the second derivatives of each function. \(g(x)=\frac{x-1}{x+1}\)

Differentiate $$ y=\frac{e^{2 x}(9 x-2)^{3}}{\sqrt[4]{\left(x^{2}+1\right)\left(3 x^{3}-7\right)}} $$

Find the first and the second derivatives of each function. \(h(s)=\frac{1}{s^{2}+2}\)

Differentiate $$ f(x)=\frac{a x}{k+x} $$

Differentiate $$ y=\frac{e^{x-1} \sin ^{2} x}{\left(x^{2}+5\right)^{2 x}} $$

Find the first and the second derivatives of each function. \(g(t)=\sqrt{3 t^{3}+2 t}\)

Differentiate $$ f(x)=\frac{a x^{2}}{4+x^{2}} $$

Find the first and the second derivatives of each function. \(f(x)=\frac{1}{x^{2}}+x-x^{3}\)

Differentiate $$ f(x)=\frac{a x^{2}}{k^{2}+x^{2}} $$

Find the first and the second derivatives of each function. \(f(s)=\sqrt{s^{3 / 2}-1}\)

Differentiate $$ f(R)=\frac{R^{n}}{k^{n}+R^{n}} $$

Find all tangent lines to the curve $$ y=x^{2} $$

Find the first and the second derivatives of each function. \(f(x)=\frac{2 x}{x^{2}+1}\)

Differentiate $$ h(t)=\sqrt{a t}(1-a)+a $$

How many tangent lines to the curve $$ y=x^{2}+2 x $$