Chapter 3

Find the derivatives of the function. $$y=\frac{1}{\left(x^{2}-1\right)\left(x^{2}+x+1\right)}$$

Find the derivative of \(y\) with respect to the appropriate variable. $$y=\csc ^{-1} \frac{x}{2}$$

A draining conical reservoir Water is flowing at the rate of \(50 \mathrm{m}^{3} / \mathrm{min}\) from a shallow concrete conical reservoir (vertex down) of base radius \(45 \mathrm{m}\) and height \(6 \mathrm{m} .\) a. How fast (centimeters per minute) is the water level falling when the water is 5 m deep? b. How fast is the radius of the water's surface changing then? Answer in centimeters per minute.

Find \(d y\). $$y=\sec \left(x^{2}-1\right)$$

Find the derivative of \(y\) with respect to \(x, t,\) or \(\theta\) as appropriate. $$y=\frac{x \ln x}{1+\ln x}$$

The number of gallons of water in a tank \(t\) minutes after the tank has started to drain is \(Q(t)=200(30-t)^{2}\) How fast is the water running out at the end of 10 min? What is the average rate at which the water flows out during the first 10 min?

Find the derivatives of the functions in Exercises \(23-50\). $$r=6(\sec \theta-\tan \theta)^{3 / 2}$$

Use implicit differentiation to find \(d y / d x\) and then \(d^{2} y / d x^{2}\) If \(x y+y^{2}=1,\) find the value of \(d^{2} y / d x^{2}\) at the point (0,-1)

Find \(d p / d q\). $$p=(1+\csc q) \cos q$$

Find an equation of the straight line having slope \(1 / 4\) that is tangent to the curve \(y=\sqrt{x}\).

Find the derivatives of the function. $$y=\frac{(x+1)(x+2)}{(x-1)(x-2)}$$

Find the derivative of \(y\) with respect to the appropriate variable. $$y=\sec ^{-1} \frac{1}{t}, \quad 0< t< 1$$

Find \(d y\). $$y=3 \csc (1-2 \sqrt{x})$$

Find the derivative of \(y\) with respect to \(x, t,\) or \(\theta\) as appropriate. $$y=\ln (\ln x)$$

Based on data from the U.S. Bureau of Public Roads, a model for the total stopping distance of a moving car in terms of its speed is $$ s=1.1 v+0.054 v^{2} $$ where \(s\) is measured in \(\mathrm{ft}\) and \(v\) in mph. The linear term \(1.1 v\) models the distance the car travels during the time the driver perceives a need to stop until the brakes are applied, and the quadratic term \(0.054 v^{2}\) models the additional braking distance once they are applied. Find \(d s / d v\) at \(v=35\) and \(v=70 \mathrm{mph},\) and interpret the meaning of the derivative.

Find the derivatives of the functions in Exercises \(23-50\). $$y=x^{2} \sin ^{4} x+x \cos ^{-2} x$$

Find \(d p / d q\). $$p=\frac{\sin q+\cos q}{\cos q}$$

An object is dropped from the top of a 100 -m-high tower. Its height above ground after \(t\) sec is \(100-4.9 t^{2} \mathrm{m} .\) How fast is it falling 2 sec after it is dropped?

Find the derivatives of the function. $$y=2 e^{-x}+e^{3 x}$$

Suppose that a drop of mist is a perfect sphere and that, through condensation, the drop picks up moisture at a rate proportional to its surface area. Show that under these circumstances the drop's radius increases at a constant rate.

Find the derivative of \(y\) with respect to the appropriate variable. $$y=\sin ^{-1} \frac{3}{t^{2}}$$

Find \(d y\). $$y=2 \cot \left(\frac{1}{\sqrt{x}}\right)$$

Find the derivative of \(y\) with respect to \(x, t,\) or \(\theta\) as appropriate. $$y=\ln (\ln (\ln x))$$

The volume \(V=(4 / 3) \pi r^{3}\) of a spherical balloon changes with the radius. a. At what rate (ft \(^{3} / \mathrm{ft}\) ) does the volume change with respect to the radius when \(r=2 \mathrm{ft} ?\) b. By approximately how much does the volume increase when the radius changes from 2 to \(2.2 \mathrm{ft} ?\)

Find the derivatives of the functions in Exercises \(23-50\). $$y=\frac{1}{x} \sin ^{-5} x-\frac{x}{3} \cos ^{3} x$$

Find the slope of the curve at the given points. $$\left(x^{2}+y^{2}\right)^{2}=(x-y)^{2} \quad \text { at } \quad(1,0) \text { and }(1,-1)$$

Find \(d p / d q\). $$p=\frac{\tan q}{1+\tan q}$$

At \(t\) sec after liftoff, the height of a rocket is \(3 t^{2} \mathrm{ft} .\) How fast is the rocket climbing \(10 \mathrm{sec}\) after liftoff?

Find the derivatives of the function. $$y=\frac{x^{2}+3 e^{x}}{2 e^{x}-x}$$

A spherical balloon is inflated with helium at the rate of \(100 \pi \mathrm{ft}^{3} / \mathrm{min}\). How fast is the balloon's radius increasing at the instant the radius is \(5 \mathrm{ft} ?\) How fast is the surface area increasing?

Find the derivative of \(y\) with respect to the appropriate variable. $$y=\cot ^{-1} \sqrt{t}$$

Find \(d y\). $$y=e^{\sqrt{x}}$$

Suppose that the distance an aircraft travels along a runway before takeoff is given by \(D=(10 / 9) t^{2},\) where \(D\) is measured in meters from the starting point and \(t\) is measured in seconds from the time the brakes are released. The aircraft will become airborne when its speed reaches \(200 \mathrm{km} / \mathrm{h}\). How long will it take to become airborne, and what distance will it travel in that time?

Find the derivatives of the functions in Exercises \(23-50\). $$y=\frac{1}{18}(3 x-2)^{6}+\left(4-\frac{1}{2 x^{2}}\right)^{-1}$$

Find \(d p / d q\). $$p=\frac{q \sin q}{q^{2}-1}$$

What is the rate of change of the area of a circle \(\left(A=\pi r^{2}\right)\) with respect to the radius when the radius is \(r=3 ?\)

Find the derivatives of the function. $$y=x^{3} e^{x}$$

Find the derivative of \(y\) with respect to the appropriate variable. $$y=\cot ^{-1} \sqrt{t-1}$$

Find \(d y\). $$y=x e^{-x}$$

Find the derivative of \(y\) with respect to \(x, t,\) or \(\theta\) as appropriate. $$y=\ln (\sec \theta+\tan \theta)$$

vent in the crater's floor, which at one point shot lava \(1900 \mathrm{ft}\) straight into the air (a Hawaiian record). What was the lava's exit velocity in feet per second? In miles per hour? (Hint: If \(v_{0}\) is the exit velocity of a particle of lava, its height \(t\) sec later will be \(s=v_{0} t-16 t^{2} \mathrm{ft} .\) Begin by finding the time at which \(d s / d t=0\) Neglect air resistance.) Although the November 1959 Kilauea Iki eruption on the island of Hawaii began with a line of fountains along the wall of the crater, activity was later confined to a single

Find the derivatives of the functions in Exercises \(23-50\). $$y=(5-2 x)^{-3}+\frac{1}{8}\left(\frac{2}{x}+1\right)^{4}$$

Verify that the given point is on the curve and find the lines that are (a) tangent and (b) normal to the curve at the given point. $$x^{2}+y^{2}=25, \quad(3,-4)$$

Find \(d p / d q\). $$p=\frac{3 q+\tan q}{q \sec q}$$

What is the rate of change of the volume of a ball \(\left(V=(4 / 3) \pi r^{3}\right)\) with respect to the radius when the radius is \(r=2 ?\)

Find the derivatives of the function. $$w=r e^{-r}$$

Find the derivative of \(y\) with respect to the appropriate variable. $$y=\ln \left(\tan ^{-1} x\right)$$

Find \(d y\). $$y=\ln \left(1+x^{2}\right)$$

Find the derivative of \(y\) with respect to \(x, t,\) or \(\theta\) as appropriate. $$y=\ln \frac{1}{x \sqrt{x+1}}$$

Give the position function \(s=f(t)\) of an object moving along the \(s\) -axis as a function of time \(t .\) Graph \(f\) together with the velocity function \(v(t)=d s / d t=f^{\prime}(t)\) and the acceleration function \(a(t)=d^{2} s / d t^{2}=f^{\prime \prime}(t) .\) Comment on the object's behavior in relation to the signs and values of \(v\) and \(a .\) Include in your commentary such topics as the following: a. When is the object momentarily at rest? b. When does it move to the left (down) or to the right (up)? c. When does it change direction? d. When does it speed up and slow down? e. When is it moving fastest (highest speed)? Slowest? f. When is it farthest from the axis origin? \(s=200 t-16 t^{2}, \quad 0 \leq t \leq 12.5\) (a heavy object fired straight up from Earth's surface at \(200 \mathrm{ft} / \mathrm{sec}\) )