Chapter 3

General logarithmic and exponential derivatives Compute the following derivatives. Use logarithmic differentiation where appropriate. $$\frac{d}{dx}(x^{\pi }+{\pi }^x)$$

Work carefully Proceed with caution when using implicit differentiation to find points at which a curve has a specified slope. For the following curves, find the points on the curve (if they exist) at which the tangent line is horizontal or vertical. Once you have found possible points, make sure they actually lie on the curve. Confirm your results with a graph. $$x^2(y-2)-e^y=0$$

A mechanical oscillator (such as a mass on a spring or a pendulum) subject to frictional forces satisfies the equation (called a differential equation) $$y^{\prime \prime }(t)+2y^{\prime }(t)+5y(t)=0,$$ where $y$ is the displacement of the oscillator from its equilibrium position. Verify by substitution that the function $y(t)=e^{-t}(\sin 2t-2\cos 2t)$ satisfies this equation.

Proof by induction: derivative of $e^{kx}$ for positive integers $k$ Proof by induction is a method in which one begins by showing that a statement, which involves positive integers, is true for a particular value (usually $k=1$ ). In the second step, the statement is assumed to be true for $k=n$, and the statement is proved for $k=n+1,$ which concludes the proof. a. Show that $\frac{d}{dx}\left(e^{kx}\right)=k{e}^{kx},$ for $k=1$ b. Assume the rule is true for $k=n$ (that is, assume $\frac{d}{dx}\left(e^{nx}\right)=n{e}^{nx}),$ and show this assumption implies that the rule is true for $k=n+1$. (Hint: Write $e^{(n+1)x}$ as the product of two functions and use the Product Rule.)

General logarithmic and exponential derivatives Compute the following derivatives. Use logarithmic differentiation where appropriate. $$\frac{d}{dx}{(1+\frac{1}{x})}^x$$

Work carefully Proceed with caution when using implicit differentiation to find points at which a curve has a specified slope. For the following curves, find the points on the curve (if they exist) at which the tangent line is horizontal or vertical. Once you have found possible points, make sure they actually lie on the curve. Confirm your results with a graph. $$x(1-y^2)+y^3=0$$

The number of hours of daylight at any point on Earth fluctuates throughout the year. In the northern hemisphere, the shortest day is on the winter solstice and the longest day is on the summer solstice. At ${40}^{\circ }$ north latitude, the length of a day is approximated by $$D(t)=12-3\cos \left(\frac{2\pi (t+10)}{365}\right),$$ where $D$ is measured in hours and $0\le t\le 365$ is measured in days, with $t=0$ corresponding to January 1 a. Approximately how much daylight is there on March 1 $(t=59)?$ b. Find the rate at which the daylight function changes. c. Find the rate at which the daylight function changes on March 1. Convert your answer to units of min/day and explain what this result means. d. Graph the function $y=D^{\prime }(t)$ using a graphing utility. e. At what times of year is the length of day changing most rapidly? Least rapidly?

Quotient Rule for the second derivative Assuming the first and second derivatives of $f$ and $g$ exist at $x,$ find a formula for $\frac{d^2}{d{x}^2}\left(\frac{f(x)}{g(x)}\right)$

A 500-liter (L) tank is filled with pure water. At time $t=0,$ a salt solution begins flowing into the tank at a rate of $5\mathrm{L}/\min .$ At the same time, the (fully mixed) solution flows out of the tank at a rate of $5.5\mathrm{L}/\min$. The mass of salt in grams in the tank at any time $t\ge 0$ is given by $$M(t)=250(1000-t)(1-{10}^{-30}(1000-t{)}^{10})$$ and the volume of solution in the tank (in liters) is given by $V(t)=500-0.5t$. a. Graph the mass function and verify that $M(0)=0$. b. Graph the volume function and verify that the tank is empty when $t=1000$ min. c. The concentration of the salt solution in the tank (in $\mathrm{g}/\mathrm{L}$ ) is given by $C(t)=M(t)/V(t).$ Graph the concentration function and comment on its properties. Specifically, what are $C(0)$ and $\underset{\theta \to {000}^{-}}{lim}C(t)?$ $t\to 1$ d. Find the rate of change of the mass $M^{\prime }(t),$ for $0\le t\le 1000$. e. Find the rate of change of the concentration $C^{\prime }(t),$ for $0\le t\le 1000$. f. For what times is the concentration of the solution increasing? Decreasing?

General logarithmic and exponential derivatives Compute the following derivatives. Use logarithmic differentiation where appropriate. $$\frac{d}{dx}{(1+x^2)}^{\sin x}$$