Chapter 4

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

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?

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

Find the tangent line, in slope-intercept form, of \(y=f(x)\) at the specified point. \(f(x)=\frac{x^{2}+3}{x^{3}+5}\), at \(x=0\)

Find a point on the curve $$ y=4-x^{2} $$ whose tangent line is parallel to the line \(y=2\). Is there more than one such point? If so, find all other points with this property.

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

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?

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

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

Find a point on the curve $$ y=(4-x)^{2} $$ whose tangent line is parallel to the line \(y=-3\). Is there more than one such point? If so, find all other points with this property.

Suppose that the concentration of nitrogen in a lake exhibits periodic behavior. That is, if we denote the concentration of nitrogen at time \(t\) by \(c(t)\), then we assume that $$c(t)=2+\sin \left(\frac{\pi}{2} t\right)$$ (a) Find \(\frac{d c}{d t}\). (b) Use a graphing calculator to graph both \(c(t)\) and \(\frac{d c}{d t}\) in the same coordinate system. (c) By inspecting the graph in (b), answer the following questions: (i) When \(c(t)\) reaches a maximum, what is the value of \(d c / d t ?\) (ii) When \(d c / d t\) is positive, is \(c(t)\) increasing or decreasing? (iii) What can you say about \(c(t)\) when \(d c / d t=0 ?\)

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?

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

Find the tangent line, in slope-intercept form, of \(y=f(x)\) at the specified point. \(f(x)=\frac{x+5}{x^{3}}\), at \(x=2\)

Find a point on the curve $$ y=2 x^{2}-\frac{1}{2} $$ whose tangent line is parallel to the line \(y=x\). Is there more than one such point? If so, find all other points with this property.

The growth rate of a fungus varies over the course of one day. You find that the size of the fungus is given as a function of time by: $$L(t)=3.6 t+1.2 \cos (2 \pi t / 24)$$ where \(t\) is the time in hours, and \(L(t)\) is the size in millimeters. (a) Calculate the growth rate \(d L / d t\) (b) What is the largest growth rate of the fungus? What is the smallest growth rate?

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

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

Find a point on the curve $$ y=1-3 x^{3} $$ whose tangent line is parallel to the line \(y=-x\). Is there more than one such point? If so, find all other points with this property.

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

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

Find a point on the curve $$ y=x^{3}+2 x+2 $$ whose tangent line is parallel to the line \(3 x-y=2\). Is there more than one such point? If so, find all other points with this property.

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

Differentiate $$ f(x)=\frac{a x}{k+x} $$ with respect to \(x\). Assume that \(a\) and \(k\) are positive constants.

Find a point on the curve $$ y=2 x^{3}-4 x+1 $$ whose tangent line is parallel to the line \(y-2 x=1\). Is there more than one such point? If so, find all other points with this property.

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

Differentiate $$ f(x)=\frac{a x^{2}}{4+x^{2}} $$ with respect to \(x\). Assume that \(a\) is a positive constant.

Show that the tangent line to the curve $$ y=x^{2} $$ at the point \((1,1)\) passes through the point \((0,-1)\).

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

Differentiate $$ f(x)=\frac{a x^{2}}{k^{2}+x^{2}} $$ with respect to \(x\). Assume that \(a\) and \(k\) are positive constants.

Find all tangent lines to the curve $$ y=x^{2} $$ that pass through the point \((0,-1)\).

Hill's function models how the amount of oxygen bound to hemoglobin in the blood depends on oxygen concentration, \(P\), in the surrounding tissues. In its most general form Hill's function models the fraction of hemoglobin molecules in blood that are bound to oxygen by: $$ f(P)=\frac{P^{n}}{k^{n}+P^{n}} $$ where \(k\) is a positive constant, and \(n\) is a positive integer. (a) Calculate \(f^{\prime}(P)\). (b) Show that \(f^{\prime}(P)>0\) for all \(P>0\). This result means that increasing the oxygen concentration always increases the fraction of hemoglobin molecules that are bound to oxygen.

Find all tangent lines to the curve $$ y=x^{2} $$ that pass through the point \(\left(0,-a^{2}\right)\), where \(a\) is a positive number.

Differentiate $$ h(t)=\sqrt{a t}(1-a)+a $$ with respect to \(t\). Assume that \(a\) is a positive constant.

How many tangent lines to the curve $$ y=x^{2}+2 x $$ pass through the point \(\left(-\frac{1}{2},-3\right) ?\)

Differentiate $$ h(t)=\sqrt{a t}(t-a)+a t $$ with respect to \(t\). Assume that \(a\) is a positive constant.

Suppose that \(f(2)=-4\), and \(f^{\prime}(2)=1\). Let \(y=1 / f(x)\); find \(\frac{d y}{d x}\) when \(x=2\).

Suppose that \(f(2)=-4, g(2)=1, f^{\prime}(2)=0\), and \(g^{\prime}(2)=-2\). Let \(y=f(x) /(2 g(x)) ;\) find \(\frac{d y}{d x}\) when \(x=2\).

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

Investigated how maximal heart rate depends on age. They found that if age \(x\) is given in years, then the maximum heart rate of a healthy adult can be predicted by the following formula: $$ H(x)=208-0.7 x $$ where \(H(x)\) is the maximum number of heart beats in one minute. The data from Tanaka et al. suggests that each additional year of age decreases \(H(x)\) by the same amount. (a) Explain in words what \(d H / d x\) represents. (b) Show that \(d H / d x\) is a constant.

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

Ultrasound is often used to make images of developing fetuses. In particular, by measuring the size of a fetus ultrasound technicians can estimate its age and then predict its birthdate. To do this requires formulas for fetus size as a function of age. Verburg et al. (2008) fit data from over 6000 fetal ultrasounds. They measured the femur length, \(L\), (in \(\mathrm{mm}\) ) as a function of the fetus age, \(t\), (in weeks) and found the following formula: $$ L=-37.50+3.71 t-6.33 \times 10^{-4} t^{3} $$ Calculate the rate of growth, \(d L / d t\), at \(t=15,20\), and 30 weeks. Does the rate of growth of the fetus increase or decrease as it ages?

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

To measure the physical properties of cells, a piezoelectric probe is used. The force applied by the probe is compared against how much the cell deforms. If \(F\) is the force applied by the probe, and \(w\) is the distance it moves into the cell, then the stiffness of the cell can be calculated from the rate of change, \(d F / d w .\) Zhang et al. found that if \(F\) is measured in \(\mu \mathrm{N}\) and \(w\) in \(\mu \mathrm{m}\) then for a zebrafish embryo: $$ F=3 \times 10^{-4} w^{3}-4.4 \times 10^{-3} w^{2}+3.93 w+0.221 $$ (a) Calculate \(d F / d w\) for this sample. (b) Stiffer cells have larger values of \(d F / d w\) when \(w=0\). Later in embryo development Zhang et al. measure: $$ F=6 \times 10^{-4} w^{3}-5.04 \times 10^{-2} w^{2}+4.08 w+1.12 $$ By calculating \(\frac{d F}{d w} \int_{w-0}\), determine whether the embryo has become stiffer or softer.

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

Assume that \(f(x)\) and \(g(x)\) are differentiable at x. Find an expression for the derivative of y in terms of \(f(x), g(x), f^{\prime}(x)\), and \(g^{\prime}(x) .\) \(y=\frac{2 f(x)+x}{3 g(x)}\)

Assume that \(f(x)\) and \(g(x)\) are differentiable at x. Find an expression for the derivative of y in terms of \(f(x), g(x), f^{\prime}(x)\), and \(g^{\prime}(x) .\) \(y=\frac{f(x)}{[g(x)]^{2}}\)

Assume that \(f(x)\) and \(g(x)\) are differentiable at x. Find an expression for the derivative of y in terms of \(f(x), g(x), f^{\prime}(x)\), and \(g^{\prime}(x) .\) \(y=\frac{x^{2}}{f(x)+g(x)}\)

Assume that \(f(x)\) and \(g(x)\) are differentiable at x. Find an expression for the derivative of y in terms of \(f(x), g(x), f^{\prime}(x)\), and \(g^{\prime}(x) .\) \(y=\sqrt{x} f(x) g(x)\)

Assume that \(f(x)\) is a differentiable function. Find the derivative of the reciprocal function \(g(x)=1 / f(x)\) at those points \(x\) where \(f(x) \neq 0\)