Chapter 4

Differentiate the functions with respect to the independent variable. $$ f(x)=\ln \frac{x}{x+1} $$

Differentiate $$ g(N)=N\left(1-\frac{N}{K}\right) $$

A measurement error in \(x\) affects the accuracy of the value \(f(x) .\) In each case, determine an interval of the form $$[f(x)-\Delta f, f(x)+\Delta f]$$ that reflects the measurement error \(\Delta x .\) In each problem, the quantities given are \(f(x)\) and \(x=\) true value of \(x \pm|\Delta x|\). $$ f(x)=\sqrt{x}, x=10 \pm 0.5 $$

The following limit represents the derivative of a function \(f\) at the point \((a, f(a))\) : $$ \lim _{h \rightarrow 0} \frac{\frac{1}{(2+h)^{2}+1}-\frac{1}{5}}{h} $$

Assume that \(f(x)\) and \(g(x)\) are differentiable. Find \(\frac{d}{d x} f\left[\frac{1}{g(x)}\right]\)

In Problems \(1-58\), find the derivative with respect to the independent variable. $$ f(x)=\tan x \cot x $$

Differentiate the functions in Problems 1-52 with respect to the independent variable. $$ f(x)=3 \sqrt{1-3 x} $$

Assume that \(f(x)\) is differentiable. Find an expression for the derivative of \(y\) at \(x=1\), assuming that \(f(1)=2\) and \(f^{\prime}(1)=-1\). $$ y=3 x^{2} f(x) $$

Differentiate the functions with respect to the independent variable. $$ f(x)=\ln \frac{2 x}{1+x^{2}} $$

Differentiate $$ g(N)=r N\left(1-\frac{N}{K}\right) $$

A measurement error in \(x\) affects the accuracy of the value \(f(x) .\) In each case, determine an interval of the form $$[f(x)-\Delta f, f(x)+\Delta f]$$ that reflects the measurement error \(\Delta x .\) In each problem, the quantities given are \(f(x)\) and \(x=\) true value of \(x \pm|\Delta x|\). $$ f(x)=e^{x}, x=2 \pm 0.2 $$

A car moves along a straight road. Its location at time \(t\) is given by $$ s(t)=20 t^{2}, 0 \leq t \leq 2 $$ where \(t\) is measured in hours and \(s(t)\) is measured in kilometers. (a) Graph \(s(t)\) for \(0 \leq t \leq 2\). (b) Find the average velocity of the car between \(t=0\) and \(t=2\). Illustrate the average velocity on the graph of \(s(t)\). (c) Use calculus to find the instantaneous velocity of the car at \(t=1\). Illustrate the instantaneous velocity on the graph of \(s(t) .\)

Assume that \(f(x)\) and \(g(x)\) are differentiable. Find \(\frac{d}{d x} \frac{[f(x)]^{2}}{g(2 x)+2 x}\)

In Problems \(1-58\), find the derivative with respect to the independent variable. $$ f(x)=\tan \left(3 x^{2}-1\right) \cot \left(3 x^{2}+1\right) $$

Differentiate the functions in Problems 1-52 with respect to the independent variable. $$ f(x)=2^{x^{2}+1} $$

Assume that \(f(x)\) is differentiable. Find an expression for the derivative of \(y\) at \(x=1\), assuming that \(f(1)=2\) and \(f^{\prime}(1)=-1\). $$ y=-5 x^{3} f(x)-2 x $$

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

Differentiate $$ g(N)=r N^{2}\left(1-\frac{N}{K}\right) $$

A measurement error in \(x\) affects the accuracy of the value \(f(x) .\) In each case, determine an interval of the form $$[f(x)-\Delta f, f(x)+\Delta f]$$ that reflects the measurement error \(\Delta x .\) In each problem, the quantities given are \(f(x)\) and \(x=\) true value of \(x \pm|\Delta x|\). $$ f(x)=\sin x, x=-1 \pm 0.05 $$

A train moves along a straight line. Its location at time \(t\) is given by $$ s(t)=\frac{100}{t}, \quad 1 \leq t \leq 5 $$ where \(t\) is measured in hours and \(s(t)\) is measured in kilometers. (a) Graph \(s(t)\) for \(1 \leq t \leq 5\). (b) Find the average velocity of the train between \(t=1\) and \(t=5\). Where on the graph of \(s(t)\) can you find the average velocity? (c) Use calculus to find the instantaneous velocity of the train at \(t=2\). Where on the graph of \(s(t)\) can you find the instantaneous velocity? What is the speed of the train at \(t=2\) ?

Find \(\frac{d y}{d x}\) by applying the chain rule repeatedly. \(y=\left(\sqrt{1-2 x^{2}}+1\right)^{2}\)

In Problems \(1-58\), find the derivative with respect to the independent variable. $$ f(x)=\sec x \cos x $$

Differentiate the functions in Problems 1-52 with respect to the independent variable. $$ f(x)=3^{x^{3}-1} $$

Assume that \(f(x)\) is differentiable. Find an expression for the derivative of \(y\) at \(x=1\), assuming that \(f(1)=2\) and \(f^{\prime}(1)=-1\). $$ y=\frac{x f(x)}{2} $$

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

Differentiate $$ g(N)=r N(a-N)\left(1-\frac{N}{K}\right) $$

Assume that the measurement of \(x\) is accurate within \(2 \% .\) In each case, determine the error \(\Delta f\) in the calculation of \(f\) and find the percentage error \(100 \frac{\Delta f}{f} .\) The quantities \(f(x)\) and the true value of \(x\) are given. $$ f(x)=4 x^{3}, x=1.5 $$

Find \(\frac{d y}{d x}\) by applying the chain rule repeatedly. \(y=\left(\sqrt{x^{3}-3 x}+3 x\right)^{4}\)

In Problems \(1-58\), find the derivative with respect to the independent variable. $$ f(x)=\sin x \sec x $$

In Problems \(41-44\), assume that \(f(x)\) and \(g(x)\) are differentiable at \(x\). Find an expression for the derivative of \(y .\) $$ y=3 f(x) g(x) $$

Differentiate the functions with respect to the independent variable. $$ f(x)=\exp [x-\ln x] $$

Differentiate $$ R(T)=\frac{2 \pi^{5}}{15} \frac{k^{4}}{c^{2} h^{3}} T^{4} $$

Assume that the measurement of \(x\) is accurate within \(2 \% .\) In each case, determine the error \(\Delta f\) in the calculation of \(f\) and find the percentage error \(100 \frac{\Delta f}{f} .\) The quantities \(f(x)\) and the true value of \(x\) are given. $$ f(x)=x^{1 / 4}, x=10 $$

4Suppose a particle moves along a straight line. The position at time \(t\) is given by $$ s(t)=3 t-t^{2}, \quad t \geq 0 $$ where \(t\) is measured in seconds and \(s(t)\) is measured in meters. (a) Graph \(s(t)\) for \(t \geq 0\). (b) Use the graph in (a) to answer the following questions: (i) Where is the particle at time \(0 ?\) (ii) Is there another time at which the particle visits the location where it was at time \(0 ?\) (iii) How far to the right on the straight line does the particle travel? (iv) How far to the left on the straight line does the particle travel? (v) Where is the velocity positive? where negative? equal to 0 ? (c) Find the velocity of the particle. (d) When is the velocity of the particle equal to \(1 \mathrm{~m} / \mathrm{s}\) ?

Find \(\frac{d y}{d x}\) by applying the chain rule repeatedly. \(y=\left(1+2(x+3)^{4}\right)^{2}\)

In Problems \(1-58\), find the derivative with respect to the independent variable. $$ f(x)=\frac{1}{\sin ^{2} x+\cos ^{2} x} $$

Differentiate the functions in Problems 1-52 with respect to the independent variable. $$ h(t)=4^{2^{3}-t} $$

Assume that \(f(x)\) and \(g(x)\) are differentiable at \(x\). Find an expression for the derivative of \(y .\) $$ y=[f(x)-3] g(x) $$

Differentiate the functions with respect to the independent variable. $$ g(s)=\exp \left[s^{2}+\ln s\right] $$

In Problems 42-48, find the tangent line, in standard form, to \(y=\) \(f(x)\) at the indicated point. $$ y=3 x^{2}-4 x+7, \text { at } x=2 $$

Assume that the measurement of \(x\) is accurate within \(2 \% .\) In each case, determine the error \(\Delta f\) in the calculation of \(f\) and find the percentage error \(100 \frac{\Delta f}{f} .\) The quantities \(f(x)\) and the true value of \(x\) are given. $$ f(x)=\ln x, x=20 $$

Find \(\frac{d y}{d x}\) by applying the chain rule repeatedly. \(y=\left(1+\left(3 x^{2}-1\right)^{3}\right)^{2}\)

In Problems \(1-58\), find the derivative with respect to the independent variable. $$ f(x)=\frac{1}{\tan ^{2} x-\sec ^{2} x} $$

Differentiate the functions in Problems 1-52 with respect to the independent variable. $$ f(x)=2^{\sqrt{x}} $$

Assume that \(f(x)\) and \(g(x)\) are differentiable at \(x\). Find an expression for the derivative of \(y .\) $$ y=[f(x)+2 g(x)] g(x) $$

Differentiate the functions with respect to the independent variable. $$ f(x)=\ln (\sin x) $$

In Problems , find the tangent line, in standard form, to \(y=\) \(f(x)\) at the indicated point. $$ y=7 x^{3}+2 x-1, \text { at } x=-3 $$

Assume that the measurement of \(x\) is accurate within \(2 \% .\) In each case, determine the error \(\Delta f\) in the calculation of \(f\) and find the percentage error \(100 \frac{\Delta f}{f} .\) The quantities \(f(x)\) and the true value of \(x\) are given. $$ f(x)=\frac{1}{1+x}, x=4 $$

Assume that \(N(t)\) denotes the size of a population at time \(t\) and that \(N(t)\) satisfies the differential equation $$ \frac{d N}{d t}=r N $$ where \(r\) is a constant. (a) Find the per capita growth rate. (b) Assume that \(r<0\) and that \(N(0)=20\). Is the population size at time 1 greater than 20 or less than \(20 ?\) Explain your answer.

Find \(\frac{d y}{d x}\) by applying the chain rule repeatedly. \(y=\left(\frac{x}{2\left(x^{2}-1\right)^{2}-1}\right)^{2}\)