Chapter 29

With Logarithmic Functions. Differentiate. $$y=\ln \left(x^{2} e^{x}\right)$$

The current in a certain \(228 \mathrm{F}\) capacitor is given by $$ i=25.5 e^{-t / 83.5} \mathrm{A} $$ Write an expression for the voltage across the capacitor when charging from an initial voltage of zero.

Tangent to a Curve Find the slope of the tangent at the point indicated. $$y=\log (4 x-3) \text { at } x=2$$

Extreme Values and Inflection Points For each curve, find the maximum, minimum, and inflection points between \(x=0\) and \(2 \pi\). $$y=3 \sin x-4 \cos x$$

With Logarithmic Functions. Differentiate. $$y=\ln x^{e}$$

The current in a certain \(3.85 \mathrm{F}\) capacitor is given by $$ i=84.6 e^{-t / 127} \mathrm{A} $$ Write an expression for the voltage across the capacitor when charging from an initial voltage of zero.

Electrical Applications The charge through a \(59.3-\Omega\) resistor is given by $$q=224 \sin (83.4 t+3.83) \quad \mathbf{C}$$ Write an expression for the instantaneous current through the resistor.

With Logarithmic Functions. Differentiate. $$y=\ln e^{2 x}$$

Angle of Intersection Find the angle of intersection of each pair of curves. $$y=\ln (x+1) \text { and } y=\ln (7-2 x) \text { at } x=2$$

Electrical Applications The voltage applied to a 22.5 -microfarad \((\mu \mathrm{F})\) capacitor is $$v=11.5 \cos (2.84 t+0.75) \quad \mathrm{V}$$ (a) Write an expression for the current in the capacitor and (b) evaluate the current at \(t=0.2 \mathrm{s}\)

Find the second derivative. $$y=e^{t} \cos t$$

Angle of Intersection Find the angle of intersection of each pair of curves. $$y=x \ln x \text { and } y=x \ln (1-x) \text { at } x=\frac{1}{2}$$

Electrical Applications The current in a 38.3 -H inductor is $$i=33.5 \sin (82.4 t+0.77) A$$ (a) Write an expression for the voltage across that inductor and (b) evaluate it at \(t=2.60 \mathrm{s}\)

Find the second derivative. $$y=e^{-t} \sin 2 t$$

Extreme Values and Points of Inflection Find the maximum, minimum, and inflection points for each curve. $$y=x \ln x$$

Find the second derivative. $$y=e^{x} \sin x$$

Find the second derivative. $$y=\frac{1}{2}\left(e^{x}+e^{-x}\right)$$

Find the smallest root that is greater than zero to two decimal places using any method. $$e^{x}+x-3=0$$

Extreme Values and Points of Inflection Find the maximum, minimum, and inflection points for each curve. $$y=\ln \left(8 x-x^{2}\right)$$

Find the smallest root that is greater than zero to two decimal places using any method. $$x e^{-0.02 x}=1$$

Find the smallest root that is greater than zero to two decimal places using any method. $$5 e^{-x}+x-5=0$$

Find the minimum point of \(y=e^{2 x}+5 e^{-2 x}\).

Find the maximum point and the points of inflection of \(y=e^{-x^{2}}\).

The pH value of a solution having a concentration \(C\) of hydrogen ions is \(\mathrm{pH}=-\log _{10} C .\) Find the rate at which the \(\mathrm{pH}\) is changing when the concentration is \(20 \times 10^{-5}\) moles/liter and decreasing at the rate of \(5.5 \times 10^{-5}\) per minute.

Find the maximum and minimum points for one cycle of \(y=10 e^{-x} \sin x\).

If 10.000 dollars is invested for \(t\) years at an annual interest rate of \(10 \%\) compounded continuously, it will accumulate to an amount \(y,\) where \(y=10,000 e^{0.1 t} .\) At what rate, in dollars per year, is the balance growing when (a) \(t=0\) years and (b) \(t=10\) years?

If we assume that the price of an automobile is increasing or "inflating" expomentially at an annual rate of \(8 \%,\) at what rate in dollars per year is the price of a car that initially cost \(\$ 9000\) increasing after 3 years?

When a certain object is placed in an oven at \(1000^{\circ} \mathrm{F}\), its temperature \(T\) rises according to the equation \(T=1000\left(1-e^{-0.1 t}\right),\) where \(t\) is the elapsed time (minutes). How fast is the temperature rising (in degrees per minute) when (a) \(t=0\) and \((b) t=10.0\) min?

A catenary has the equation \(y=\frac{1}{2}\left(e^{x}+e^{-x}\right) .\) We have seen the catenary before. It is the shape taken by a rope or chain suspended from both ends. Find the slope of the catenary when \(x=5\).

The speed \(N\) of a certain flywheel is decaying exponentially according to the equation \(N=1855 e^{-0.5 t}\) (rev/min), where \(t\) is the time (min) after the power is disconnected. Find the angular acceleration (the rate of change of \(N\) ) when \(t=1 \mathrm{min}\).

The height \(y\) of a certain pendulum released from a height of \(50.0 \mathrm{cm}\) is \(y=50.0 e^{-0.5 t} \mathrm{cm},\) where \(t\) is the time after release in seconds. Find the vertical component of the velocity of the pendulum when \(t=1.00 \mathrm{s}\).

The atmospheric pressure at a height of \(h\) miles above the earth's surface is given by \(p=29.92 e^{-h / 5}\) in. of mercury. Find the rate of change of the pressure on a rocket that is at 18.0 mi and climbing at a rate of \(1500 \mathrm{mi} / \mathrm{h}\).

The equation in problem 59 becomes \(p=2121 e^{-0.000037 h}\) when \(h\) is in feet and \(p\) is in pounds per square foot. Find the rate of change of pressure on an aircraft at \(5000 \mathrm{ft}\) climbing at a rate of \(10 \mathrm{ft} / \mathrm{s}\).

The approximate density of seawater at a depth of \(h\) miles is \(d=64.0 e^{0.00676 h} \mathrm{lb} / \mathrm{ft}^{3} .\) Find the rate of change of density, with respect to depth, at a depth of 1.00 mile.

The voltage applied to a certain 218 -microfarad \((\mu \mathrm{F})\) capacitor is $$ v=25.4\left(1-e^{-t / 285}\right) \quad \mathbf{V} $$ (a) Write an expression for the current in the capacitor and (b) evaluate the current at \(t=200 \mathrm{s}\).

The voltage applied to a certain 185 -microfarad \((\mu \mathrm{F})\) capacitor is $$v=448\left(1-e^{-t / 122}\right) \quad \mathbf{v}$$ (a) Write an expression for the current in the capacitor and (b) evaluate the current at \(t=150 \mathrm{s}\).

The current in a \(88.3-\mathrm{H}\) inductor is $$i=115 e^{-t / 624} \mathrm{A}$$ (a) Write an expression for the voltage across that inductor and (b) evaluate it at \(t=250 \mathrm{s}\).

The current in a \(37.2-\mathrm{H}\) inductor is $$i=225 e^{-t / 128} \mathrm{A}$$ (a) Write an expression for the voltage across that inductor and (b) evaluate it at \(t=155 \mathrm{s}\)