Chapter 4

Use the identity $$\cos (\alpha+\beta)=\cos \alpha \cos \beta-\sin \alpha \sin \beta$$ and the definition of the derivative to show that $$ \frac{d}{d x} \cos x=-\sin x $$

Suppose that a bacterial colony grows in such a way that at time \(t\) the population size is $$ N(t)=N_{0} 2^{t} $$ where \(N_{0}\) is the population size at time \(0 .\) Find the rate of growth \(d N / d t .\) Express your solution in terms of \(N(t) .\) Show that the growth rate of the population is proportional to the population size.

Let \(f(x)=\ln x\). We know that \(f^{\prime}(x)=\frac{1}{x}\). We will use th fact and the definition of derivatives to show that $$ \lim _{n \rightarrow \infty}\left(1+\frac{1}{n}\right)^{n}=e $$ (a) Use the definition of the derivative to show that $$ f^{\prime}(1)=\lim _{h \rightarrow 0} \frac{\ln (1+h)}{h} $$ (b) Show that (a) implies that $$ \ln \left[\lim _{h \rightarrow 0}(1+h)^{1 / h}\right]=1 $$ (c) Set \(h=\frac{1}{n}\) in (b) and let \(n \rightarrow \infty\). Show that this implies that $$ \lim _{n \rightarrow \infty}\left(1+\frac{1}{n}\right)^{n}=e $$

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

Find the normal line to $$ f(x)=\frac{a x^{2}}{a+1} $$ at \(x=2 .\) Assume that \(a\) is a positive constant.

Use the quotient rule to show that $$ \frac{d}{d x} \cot x=-\csc ^{2} x $$

Suppose that a bacterial colony grows in such a way that at time \(t\) the population size is $$ N(t)=N_{0} 2^{t} $$ where \(N_{0}\) is the population size at time \(0 .\) Find the per capita growth rate. $$ \frac{1}{N} \frac{d N}{d t} $$

Assume that \(f(x)\) is differentiable with respect to \(x\). Show that $$ \frac{d}{d x} \ln \left[\frac{f(x)}{x}\right]=\frac{f^{\prime}(x)}{f(x)}-\frac{1}{x} $$

Differentiate with respect to the independent variable. \(f(x)=\frac{\sqrt{5 x}\left(1+x^{2}\right)}{\sqrt{2}}\)

Find the normal line to $$ f(x)=\frac{x^{3}}{a+1} $$ at \(x=2 a\). Assume that \(a\) is a positive constant.

Use the quotient rule to show that $$ \frac{d}{d x} \sec x=\sec x \tan x $$

A population of organisms grows according to a logistic growth model: $$ N(t)=\frac{K}{1+(K-1) e^{-r t}} $$ where \(r\) and \(K\) are positive constants. (a) Find \(\frac{d N}{d t}\). (b) Show that \(N(t)\) satisfies the equation $$ \frac{d N}{d t}=r N\left(1-\frac{N}{K}\right) $$ [Hint: Use the function \(N(t)\) for the right-hand side, and simplify until you obtain the derivative of \(N(t)\) that you computed in (a).] (c) Plot the per capita rate of growth \(\frac{1}{N} \frac{d N}{d t}\) as a function of \(N\), and note that it decreases with increasing population size.

Royston and Wright (1998) studied how the size of an unborn fetus depends on its age. They fitted data for head circumference \((H)\) as a function of age \((t)\) in weeks using the formula $$ H=-29.32+1.705 t^{2}-0.3981 t^{2} \log t $$ (a) Calculate the rate of fetal growth \(d H / d t\). (b) Is \(d H / d t\) larger early in development (say at \(t=8\) weeks) or late (say at \(t=36\) weeks \()\) ? (c) Repeat part (b) but for fractional rate of growth \(\frac{1}{H} \frac{d H}{d t}\).

Differentiate with respect to the independent variable. \(f(x)=x^{3}+\frac{1}{x^{3}}\)

In Problems 63-70, find the coordinates of all of the points of the graph of \(y=f(x)\) that have horizontal tangents. $$ f(x)=x^{2} $$

Use the quotient rule to show that $$ \frac{d}{d x} \csc x=-\csc x \cot x $$

The following model is used in the fisheries literature to describe the recruitment of fish as a function of the size of the parent stock: If we denote the number of recruits by \(R\) and the size of the parent stock by \(P\), then $$ R(P)=a P e^{-b P}, \quad P \geq 0 $$ where \(a\) and \(b\) are positive constants. (a) Sketch the graph of the function \(R(P)\) when \(b=1\) and \(a=2\). (b) Differentiate \(R(P)\) with respect to \(P .\) (c) Find all the points on the curve that have a horizontal tangent.

Some worms swim by passing an undulatory wave along their bodies. The force that small worms apply to the water by passing this wave can be modeled using a formula derived by Lamb (1911) $$ F=\frac{4 \pi L \mu U}{\left(-0.077-\ln \left(\frac{\rho U a}{4 \mu}\right)\right)} $$ where \(U\) is the velocity of undulation, \(L\) is the length of the worm, \(a\) is the radius of the worm's body, and \(\mu\) and \(\rho\) respectively the viscosity (or "stickiness") and density of the water through which the worm swims. Calculate \(d F / d U\), the rate of change of the force with increasing undulation velocity.

Differentiate with respect to the independent variable. \(f(x)=x^{5}-\frac{1}{x^{5}}\)

Find the coordinates of all of the points of the graph of \(y=f(x)\) that have horizontal tangents. $$ f(x)=2-x^{2} $$

Find the derivatives of the following functions: $$ f(x)=\sin \sqrt{x^{2}+1} $$

Von Bertalanffy Growth Model The growth of fish can be described by the von Bertalanffy growth function $$ L(x)=L_{\infty}-\left(L_{\infty}-L_{0}\right) e^{-k x} $$ where \(x\) denotes the age of the fish and \(k, L_{\infty}\), and \(L_{0}\) are positive constants. (a) \(\operatorname{Set} L_{0}=1\) and \(L_{\infty}=10 .\) Graph \(L(x)\) for \(k=1.0\) and \(k=0.1\). (b) Interpret \(L_{\infty}\) and \(L_{0}\). (c) Compare the graphs for \(k=0.1\) and \(k=1.0 .\) According to your graphs which fish reach \(L=5\) more quickly? (d) Show that $$ \frac{d}{d x} L(x)=k\left(L_{\infty}-L(x)\right) $$ That is, \(d L / d x \propto L_{\infty}-L .\) What does this proportionality say about how the rate of growth changes with age? (e) The constant \(k\) is the proportionality constant in (d). What does the value of \(k\) tell you about how quickly a fish grows?

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

Differentiate with respect to the independent variable. \(f(x)=2 x^{2}-\frac{x-3}{x^{2}}\)

Find the coordinates of all of the points of the graph of \(y=f(x)\) that have horizontal tangents. $$ f(x)=3 x-x^{2} $$

Find the derivatives of the following functions: $$ f(x)=\cos \sqrt{x^{2}+1} $$

Radioactive Decay Suppose \(W(t)\) denotes the amount of a radioactive material left after time \(t\) (measured in days). Assume that the radioactive decay rate of the material is \(0.2 /\) day. Find the differential equation for the radioactive decay function \(W(t)\).

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

Differentiate with respect to the independent variable. \(f(x)=-x^{3}+\frac{2 x^{2}+3}{x^{4}+1}\)

Find the coordinates of all of the points of the graph of \(y=f(x)\) that have horizontal tangents. $$ f(x)=4 x+2 x^{2} $$

Find the derivatives of the following functions: $$ f(x)=\sin \sqrt{3 x^{2}+3 x} $$

Radioactive Decay Suppose \(W(t)\) denotes the amount of a radioactive material left after time \(t\) (measured in days). Assume that the radioactive decay rate of the material is \(4 /\) day. Find the differential equation for the radioactive decay function \(W(t)\).

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

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

Find the coordinates of all of the points of the graph of \(y=f(x)\) that have horizontal tangents. $$ f(x)=3 x^{3}-x^{2} $$

Find the derivatives of the following functions: $$ f(x)=\cos x-\sin 2 x $$

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 \(W(t)\).

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

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

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

Find the derivatives of the following functions: $$ f(x)=\sin ^{2}\left(x^{2}-1\right) $$

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 \(W(t)\).

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

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

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

Find the derivatives of the following functions: $$ f(x)=\cos ^{2}\left(2 x^{2}+3\right) $$

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?

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

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

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