Chapter 4

Calculate the linear approximation for \(f(x)\) : $$f(x) \approx f(a)+f^{\prime}(a)(x-a)$$ \(f(x)=(1-x)^{-n}\) at \(a=0 .\) (Assume that \(n\) is a positive integer.)

Find the equation of the tangent line to the curve \(y=2 / x\) at the point \((2,1)\).

Find the derivative with respect to the independent variable. $$ f(x)=\sqrt{\sin x} $$

Differentiate the functions with respect to the independent variable. (Note that log denotes the logarithm to base 10.) $$ f(x)=\ln \left(1-x^{2}\right) $$

Apply the product rule for the product of three functions to find the derivative of \(y=f(x)\). \(f(x)=(2 x+1)\left(4-x^{2}\right)\left(1+x^{2}\right)\)

Differentiate $$ f(r)=r s^{2}-r $$ with respect to \(r\). Assume that \(s\) is a constant.

In Problems , graph each function and, on the basis of the graph, guess where the function is not differentiable. (Assume the largest possible domain.) $$ y=|x+2|-1 $$

Tree Growth Rate Sperry et al. (2012) studied how the growth rates \(G\) of trees depend upon their body mass, \(M\). They argued that \(G=C M^{0.7}\) for some constant \(C .\) As the tree grows, \(M\) changes. (a) Show how \(d G / d t\) is related to \(d M / d t\). (b) Show how the fractional rate of increase of \(G, \frac{1}{G} \frac{d G}{d t}\), is related to the fractional rate of growth \(\frac{1}{M} \frac{d M}{d t}\).

In Problems 1-28, differentiate the functions with respect to the independent variable. $$ h(t)=\left(4 t^{4}+\frac{4}{t^{4}}\right)^{1 / 4} $$

Calculate the linear approximation for \(f(x)\) : $$f(x) \approx f(a)+f^{\prime}(a)(x-a)$$ \(f(x)=\sqrt{1+x^{2}}\) at \(a=0\)

Find the equation of the tangent line to the curve \(y=\sqrt{x}\) at the point \((4,2)\).

Find the derivative with respect to the independent variable. $$ f(x)=\sqrt{\sin \left(2 x^{2}-1\right)} $$

Differentiate the functions with respect to the independent variable. (Note that log denotes the logarithm to base 10.) $$ f(x)=\ln \left(2 x^{3}-x\right) $$

Differentiate $$ f(x)=a(x+1)(2 x-1) $$ with respect to \(x\). Assume that \(a\) is a positive constant.

Differentiate $$ f(x)=r s^{2} x^{3}-r x+s $$ with respect to \(x\). Assume that \(r\) and \(s\) are constants.

In Problems , graph each function and, on the basis of the graph, guess where the function is not differentiable. (Assume the largest possible domain.)$$ y=2-|x-3| $$

Differentiate $$ f(x)=(a x+1)^{3} $$ with respect to \(x\). Assume that \(a\) is a positive constant.

Calculate the linear approximation for \(f(x)\) : $$f(x) \approx f(a)+f^{\prime}(a)(x-a)$$ \(f(x)=\left(1+\frac{1}{x}\right)^{1 / 4}\) at \(a=1\)

Find the equation of the tangent line to the curve \(y=x^{2}-\) \(3 x+1\) at the point \((2,-1)\)

Find the derivative with respect to the independent variable. $$ g(s)=\left(\cos ^{2} s-3 s^{2}\right)^{2} $$

Differentiate the functions with respect to the independent variable. (Note that log denotes the logarithm to base 10.) $$ f(x)=\ln \left(1-x^{3}\right) $$

Differentiate $$ f(x)=(a-x)(a+x) $$ with respect to \(x\). Assume that \(a\) is a positive constant.

Differentiate $$ f(x)=\frac{r+x}{r s^{2}}-r s x+(r+s) x-r s $$ with respect to \(x\). Assume that \(r\) and \(s\) are nonzero constants.

In Problems , graph each function and, on the basis of the graph, guess where the function is not differentiable. (Assume the largest possible domain.)$$ y=\frac{1}{2+x} $$

Allometric Equations Suppose that two quantities, \(y\) and \(x\) are related by a power law: $$ y=k x^{a} $$ where \(k\) and \(a\) are both constants. \(x\) grows with time at a rate \(d x / d t\) (a) Explain why \(\frac{1}{x} \frac{d x}{d t}\) can be thought of as the relative rate of growth of \(x\). (b) Show that the relative rates of growth of \(y\) and \(x\) are related by an equation: $$ \frac{1}{y} \frac{d y}{d t}=\frac{a}{x} \frac{d x}{d t} $$

Differentiate $$ f(x)=\sqrt{a x^{2}-2} $$ with respect to \(x\). Assume that \(a\) is a positive constant.

Suppose that the per capita growth rate of a population is \(3 \%\); that is, if \(N(t)\) denotes the population size at time \(t\), then $$\frac{1}{N} \frac{d N}{d t}=0.03$$ Suppose also that the population size at time \(t=4\) is equal to 100 . Use a linear approximation to compute the population size at time \(t=4.1\).

Find the equation of the normal line to the curve \(y=-3 x^{2}+x\) at the point \((-1,-4)\).

Differentiate the functions with respect to the independent variable. (Note that log denotes the logarithm to base 10.) $$ f(x)=(\ln x)^{2} $$

Differentiate $$ f(x)=2 a\left(x^{2}-a\right)+a^{2} $$ with respect to \(x\). Assume that \(a\) is a positive constant.

Differentiate $$ f(N)=(b-1) N^{4}-\frac{N^{2}}{b} $$ with respect to \(N\). Assume that \(b\) is a nonzero constant.

Differentiate $$ g(N)=\frac{b N}{k+N} $$ with respect to \(N\). Assume that \(b\) and \(k\) are positive constants.

Suppose that the per capita growth rate of a population is \(2 \% ;\) that is, if \(N(t)\) denotes the population size at time \(t\), then $$\frac{1}{N} \frac{d N}{d t}=0.02$$ Suppose also that the population size at time \(t=2\) is equal to \(100 .\) Use a linear approximation to compute the population size at time \(t=2.1\).

Find the equation of the normal line to the curve \(y=4 / x\) at the point \((-1,-4)\).

Find the derivative with respect to the independent variable. $$ g(t)=\frac{\sin (3 t)}{\cos (5 t)} $$

Differentiate the functions with respect to the independent variable. (Note that log denotes the logarithm to base 10.) $$ f(x)=(\ln x)^{3} $$

Differentiate $$ f(x)=\frac{3(x-1)^{2}}{2+a} $$ with respect to \(x\). Assume that \(a\) is a positive constant.

Differentiate $$ f(N)=\frac{b N^{2}+N}{K+b} $$ with respect to \(N\). Assume that \(b\) and \(K\) are positive constants.

In Problems , graph each function and, on the basis of the graph, guess where the function is not differentiable. (Assume the largest possible domain.)$$ y=\frac{x-1}{x+1} $$

Differentiate $$ g(N)=\frac{N}{(k+b N)^{2}} $$ with respect to \(N\). Assume that \(b\) and \(k\) are positive constants.

Suppose that the specific growth rate of a plant is \(1 \%\); that is, if \(B(t)\) denotes the biomass at time \(t\), then $$\frac{1}{B(t)} \frac{d B}{d t}=0.01$$ Suppose that the biomass at time \(t=1\) is equal to 5 grams. Use a linear approximation to compute the biomass at time \(t=1.1\).

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

Find the derivative with respect to the independent variable. $$ g(t)=\frac{\sin (2 t)+1}{\cos (6 t)-1} $$

Differentiate the functions with respect to the independent variable. (Note that log denotes the logarithm to base 10.) $$ f(x)=\left(\ln x^{2}\right)^{2} $$

Differentiate $$ g(t)=(a t+1)^{2} $$ with respect to \(t\). Assume that \(a\) is a positive constant.

Differentiate $$ g(t)=a^{3} t-a t^{3} $$ with respect to \(t\). Assume that \(a\) is a constant.

In Problems , graph each function and, on the basis of the graph, guess where the function is not differentiable. (Assume the largest possible domain.)$$ y=\frac{3-x}{3+x} $$

Differentiate $$ g(T)=a\left(T_{0}-T\right)^{3}-b $$ with respect to \(T\). Assume that \(a, b\), and \(T_{0}\) are positive constants.

Suppose that a certain plant is grown along a gradient ranging from nitrogen- poor to nitrogen-rich soil. Experimental data show that the average mass per plant grown in a soil with a total nitrogen content of \(1000 \mathrm{mg}\) nitrogen per \(\mathrm{kg}\) of soil is \(2.7 \mathrm{~g}\) and the rate of change of the average mass per plant at this nitrogen level is \(1.05 \times 10^{-3} \mathrm{~g}\) per mg change in total nitrogen per \(\mathrm{kg}\) soil. Use a linear approximation to predict the average mass per plant grown in a soil with a total nitrogen content of \(1100 \mathrm{mg}\) nitrogen per \(\mathrm{kg}\) of soil.

Find the equation of the normal line to the curve \(y=\sqrt{x}-1\) at the point \((4,1)\).