Chapter 4

\(37-50=\) Find the absolute maximum and absolute minimum values of \(f\) on the given interval. $$f(x)=\left(x^{2}-1\right)^{3}, \quad[-1,2]$$

What is the smallest possible area of the triangle that is cut off by the first quadrant and whose hypotenuse is tangent to the parabola \(y=4-x^{2}\) at some point?

(a) Find the vertical and horizontal asymptotes. (b) Find the intervals of increase or decrease. (c) Find the local maximum and minimum values. (d) Find the intervals of concavity and the inflection points. (e) Use the information from parts ( d ) to sketch the graph of \(f .\) \(f(x)=x-\frac{1}{6} x^{2}-\frac{2}{3} \ln x\)

A stone is dropped from the upper observation deck (the Space Deck) of the CN Tower, 450 \(\mathrm{m}\) above the ground. (a) Find the distance of the stone above ground level at time \(t .\) (b) How long does it take the stone to reach the ground? (c) With what velocity does it strike the ground? (d) If the stone is thrown downward with a speed of \(5 \mathrm{m} / \mathrm{s},\) how long does it take to reach the ground?

\(37-50=\) Find the absolute maximum and absolute minimum values of \(f\) on the given interval. $$f(t)=t \sqrt{4-t^{2}}, \quad[-1,2]$$

(a) If \(C(x)\) is the cost of producing \(x\) units of a commodity, then the average cost per unit is \(c(x)=C(x) / x\) . Show that if the average cost is a minimum, then the marginal cost equals the average cost. (b) If \(C(x)=16,000+200 x+4 x^{3 / 2},\) in dollars, find (i) the cost, average cost, and marginal cost at a pro- duction level of 1000 units; (ii) the production level that will minimize the average cost; and (iii) the mini- mum average cost.

(a) Find the vertical and horizontal asymptotes. (b) Find the intervals of increase or decrease. (c) Find the local maximum and minimum values. (d) Find the intervals of concavity and the inflection points. (e) Use the information from parts ( d ) to sketch the graph of \(f .\) \(f(x)=\ln (1-\ln x)\)

Show that for motion in a straight line with constant acceleration \(a\) , initial velocity \(v_{0},\) and initial displacement \(s_{0},\) the displacement after time \(t\) is \(s=\frac{1}{2} a t^{2}+v_{0} t+s_{0}\)

\(37-50=\) Find the absolute maximum and absolute minimum values of \(f\) on the given interval. $$f(x)=\frac{x}{x^{2}-x+1},[0,3]$$

(a) Show that if the profit \(P(x)\) is a maximum, then the marginal revenue equals the marginal cost. (b) If \(C(x)=16,000+500 x-1.6 x^{2}+0.004 x^{3}\) is the cost function and \(p(x)=1700-7 x\) is the demand function, find the production level that will maximize profit.

(a) Find the vertical and horizontal asymptotes. (b) Find the intervals of increase or decrease. (c) Find the local maximum and minimum values. (d) Find the intervals of concavity and the inflection points. (e) Use the information from parts ( d ) to sketch the graph of \(f .\) \(f(x)=e^{\arctan x}\)

\(37-50=\) Find the absolute maximum and absolute minimum values of \(f\) on the given interval. $$f(t)=2 \cos t+\sin 2 t, \quad[0, \pi / 2]$$

In the theory of relativity, the mass of a particle is $$m=\frac{m_{0}}{\sqrt{1-v^{2} / c^{2}}}$$ where \(m_{0}\) is the rest mass of the particle, \(m\) is the mass when the particle moves with speed \(v\) relative to the observer, and \(c\) is the speed of light. Sketch the graph of \(m\) as a function of \(v .\)

A baseball team plays in a stadium that holds \(55,000\) spectators. With ticket prices at \(\$ 10,\) the average attendance had been \(27,000 .\) When ticket prices were lowered to \(\$ 8,\) the average attendance rose to \(33,000\) . (a) Find the demand function, assuming that it is linear. (b) How should ticket prices be set to maximize revenue?

Suppose the derivative of a function \(f\) is \(f^{\prime}(x)=(x+1)^{2}(x-3)^{5}(x-6)^{4} .\) On what interval is \(f\) increasing?

\(37-50=\) Find the absolute maximum and absolute minimum values of \(f\) on the given interval. $$f(t)=t+\cot \left(\frac{1}{2} t\right), \quad[\pi / 4,7 \pi / 4]$$

During the summer months Terry makes and sclls necklaces on the beach. Last summer he sold the necklaces for \(\$ 10\) each and his sales averaged 20 per day. When he increased the price by \(\$ 1,\) he found that the average decreased by two sales per day. (a) Find the demand function, assuming that it is linear. (b) If the material for each necklace costs Terry \(\$ 6,\) what should the selling price be to maximize his profit?

In the theory of relativity, the energy of a particle is $$E=\sqrt{m_{0}^{2} c^{4}+h^{2} c^{2} / \lambda^{2}}$$ where \(m_{0}\) is the rest mass of the particle, \(\lambda\) is its wave length, and \(h\) is Planck's constant. Sketch the graph of \(E\) as a function of \(\lambda .\) What does the graph say about the energy?

Use the methods of this section to sketch the curve \(y=x^{3}-3 a^{2} x+2 a^{3},\) where \(a\) is a positive constant. What do the members of this family of curves have in common? How do they differ from each other?

A stone was dropped off a cliff and hit the ground with a speed of 120 \(\mathrm{ft} / \mathrm{s} .\) What is the height of the cliff?

\(37-50=\) Find the absolute maximum and absolute minimum values of \(f\) on the given interval. $$f(x)=x e^{-x^{2} / 8}, \quad[-1,4]$$

A manufacturer has been selling 1000 flat-screen TVs a weck at \(\$ 450\) cach. A market survey indicates that for cach \(\$ 10\) rebate offered to the buyer, the number of TVs sold will increase by 100 per week. (a) Find the demand function. (b) How large a rebate should the company offer the buyer in order to maximize its revenue? (c) If its weekly cost function is \(C(x)=68,000+150 x\) how should the manufacturer set the size of the rebate in order to maximize its profit?

If a diver of mass \(m\) stands at the end of a diving board with length \(L\) and linear density \(\rho,\) then the board takes on the shape of a curve \(y=f(x),\) where \(\quad E I y^{\prime \prime}=m g(L-x)+\frac{1}{2} \rho g(L-x)^{2}\) \(E\) and \(I\) are positive constants that depend on the material of the board and \(g(<0)\) is the acceleration due to gravity. (a) Find an expression for the shape of the curve. (b) Use \(f(L)\) to estimate the distance below the horizontal at \(\quad\) the end of the board.

\(37-50=\) Find the absolute maximum and absolute minimum values of \(f\) on the given interval. $$f(x)=x-\ln x, \quad\left[\frac{1}{2}, 2\right]$$

Coulomb's Law states that the force of attraction between two charged particles is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. The figure shows particles with charge 1 located at positions 0 and 2 on a coordinate line and a particle with charge \(-1\) at a position \(x\) between them. It follows from Coulomb's Law that the net force acting on the middle particle is $$F(x)=-\frac{k}{x^{2}}+\frac{k}{(x-2)^{2}} \quad 0< x <2$$ where \(k\) is a positive constant. Sketch the graph of the net force function. What does the graph say about the force?

The manager of a 100 -unit apartment complex knows from experience that all units will be occupied if the rent is \(\$ 800\) per month. A market survey suggests that, on average, one additional unit will remain vacant for each \(\$ 10\) increase in rent. What rent should the manager charge to maximize revenue?

since raindrops grow as they fall, their surface area increases and therefore the resistance to their falling increases. A raindrop has an initial downward velocity of 10 \(\mathrm{m} / \mathrm{s}\) and its downward acceleration is $$a=\left\\{\begin{array}{ll}{9-0.9 t} & {\text { if } 0 \leqslant t \leqslant 10} \\ {0} & {\text { if } t>10}\end{array}\right.$$ If the raindrop is initially 500 \(\mathrm{m}\) above the ground, how long does it take to fall?

\(37-50=\) Find the absolute maximum and absolute minimum values of \(f\) on the given interval. $$f(x)=\ln \left(x^{2}+x+1\right), \quad[-1,1]$$

\(49-52=\) The line \(y=m x+b\) is called a slant asymptote if \(f(x)-(m x+b) \rightarrow 0\) as \(x \rightarrow \infty\) or \(x \rightarrow-\infty\) because the vertical distance between the curve \(y=f(x)\) and the line \(y=m x+b\) approaches 0 as \(x\) becomes large. Find an equation of the slant asymptote of the function and use it to help sketch the graph. [ For rational functions, a slant asymptote occurs when the degree of the numerator is one more than the degree of the denominator. To find it, use long division to write $$f(x)=m x+b+R(x) / Q(x) ]$$ $$y=\frac{x^{2}}{x-1}$$

Let \(a\) and \(b\) be positive numbers. Find the length of the shortest line segment that is cut off by the first quadrant and passes through the point \((a, b) .\)

A drug response curve describes the level of medication in the bloodstream after a drug is administered. A surge function \(S(t)=A t^{p} e^{-k t}\) is often used to model the response curve, reflecting an initial surge in the drug level and then a more gradual decline. If, for a particular drug,A \(=0.01\) \(p=4, k=0.07,\) and \(t\) is measured in minutes, estimate the times corresponding to the inflection points and explain their significance. If you have a graphing device, use it to graph the drug response curve.

A car is traveling at 50 \(\mathrm{mi} / \mathrm{h}\) when the brakes are fully applied, producing a constant deceleration of 22 \(\mathrm{ft} / \mathrm{s}^{2} .\) What is the distance traveled before the car comes to a stop?

\(37-50=\) Find the absolute maximum and absolute minimum values of \(f\) on the given interval. $$f(x)=x-2 \tan ^{-1} x, \quad[0,4]$$

\(49-52=\) The line \(y=m x+b\) is called a slant asymptote if \(f(x)-(m x+b) \rightarrow 0\) as \(x \rightarrow \infty\) or \(x \rightarrow-\infty\) because the vertical distance between the curve \(y=f(x)\) and the line \(y=m x+b\) approaches 0 as \(x\) becomes large. Find an equation of the slant asymptote of the function and use it to help sketch the graph. [ For rational functions, a slant asymptote occurs when the degree of the numerator is one more than the degree of the denominator. To find it, use long division to write $$f(x)=m x+b+R(x) / Q(x) ]$$ $$y=\frac{1+5 x-2 x^{2}}{x-2}$$

The family of bell-shaped curves $$y=\frac{1}{\sigma \sqrt{2 \pi}} e^{-(x-\mu)^{2} /\left(2 \sigma^{2}\right)}$$ occurs in probability and statistics, where it is called the normal density function. The constant \(\mu\) is called the mean and the positive constant \(\sigma\) is called the standard deviation. For simplicity, let's scale the function so as to remove the factor 1\(/(\sigma \sqrt{2 \pi})\) and let's analyze the special case where \(\mu=0 .\) So we study the function $$f(x)=e^{-x^{2} /\left(2 \sigma^{2}\right)}$$ (a) Find the asymptote, maximum value, and inflection points of \(f .\) (b) What role does \(\sigma\) play in the shape of the curve? (c) Illustrate by graphing four members of this family on the same screen.

What constant acceleration is required to increase the speed of a car from 30 \(\mathrm{mi} / \mathrm{h}\) to 50 \(\mathrm{mi} / \mathrm{h}\) in 5 \(\mathrm{s} ?\)

If \(a\) and \(b\) are positive numbers, find the maximum value of \(f(x)=x^{a}(1-x)^{b}, 0 \leqslant x \leqslant 1\)

\(49-52=\) The line \(y=m x+b\) is called a slant asymptote if \(f(x)-(m x+b) \rightarrow 0\) as \(x \rightarrow \infty\) or \(x \rightarrow-\infty\) because the vertical distance between the curve \(y=f(x)\) and the line \(y=m x+b\) approaches 0 as \(x\) becomes large. Find an equation of the slant asymptote of the function and use it to help sketch the graph. [ For rational functions, a slant asymptote occurs when the degree of the numerator is one more than the degree of the denominator. To find it, use long division to write $$f(x)=m x+b+R(x) / Q(x) ]$$ $$y=\frac{x^{3}+4}{x^{2}}$$

Let \(v_{1}\) be the velocity of light in air and \(v_{2}\) the velocity of light in water. According to Fermat's Principle, a ray of light will travel from a point \(A\) in the air to a point \(B\) in the water by a path \(A C B\) that minimizes the time taken. Show that $$\frac{\sin \theta_{1}}{\sin \theta_{2}}=\frac{v_{1}}{v_{2}} $$ where \(\theta_{1}\) (the angle of incidence) and \(\theta_{2}\) (the angle of refraction) are as shown. This cquation is known as Snell's Law.

Find a cubic function \(f(x)=a x^{3}+b x^{2}+c x+d\) that has a local maximum value of 3 at \(x=-2\) and a local minimum value of 0 at \(x=1.\)

A car braked with a constant deceleration of \(16 \mathrm{ft} / \mathrm{s}^{2},\) pro- ducing skid marks measuring 200 \(\mathrm{ft}\) before coming to a stop. How fast was the car traveling when the brakes were first applied?

For what values of the numbers \(a\) and \(b\) does the function $$f(x)=a x e^{b x^{2}}$$ have the maximum value \(f(2)=1 ?\)

A car is traveling at 100 \(\mathrm{km} / \mathrm{h}\) when the driver sees an accident 80 \(\mathrm{m}\) ahead and slams on the brakes. What constant deceleration is required to stop the car in time to avoid a pileup?

\(53-56\) (a) Use a graph to estimate the absolute maximum and minimum values of the function to two decimal places. (b) Use calculus to find the exact maximum and minimum values. $$f(x)=x^{5}-x^{3}+2, \quad-1 \leqslant x \leqslant 1$$

Show that the curve \(y=x-\tan ^{-1} x\) has two slant asymptotes: \(y=x+\pi / 2\) and \(y=x-\pi / 2 .\) Use this fact to help sketch the curve.

(a) If the function \(f(x)=x^{3}+a x^{2}+b x\) has the local minimum value \(-\frac{2}{9} \sqrt{3}\) at \(x=1 / \sqrt{3},\) what are the values of \(a\) and \(b ?\) (b) Which of the tangent lines to the curve in part (a) has the smallest slope?

A model rocket is fired vertically upward from rest. Its acceleration for the first three seconds is \(a(t)=60 t,\) at which time the fuel is exhausted and it becomes a freely "falling" body. Fourteen seconds later, the rocket's parachute opens, and the (downward) velocity slows linearly to \(-18 \mathrm{ft} / \mathrm{s}\) in 5 \(\mathrm{s}\) . The rocket then "floats" to the ground at that rate. (a) Determine the position function \(s\) and the velocity function \(v(\) for all times \(t) .\) Sketch the graphs of \(s\) and \(v .\) (b) At what time does the rocket reach its maximum height, and what is that height? (c) At what time does the rocket land?

\(53-56\) (a) Use a graph to estimate the absolute maximum and minimum values of the function to two decimal places. (b) Use calculus to find the exact maximum and minimum values. $$f(x)=e^{x}+e^{-2 x}, \quad 0 \leqslant x \leqslant 1$$

Show that the curve \(y=\sqrt{x^{2}+4 x}\) has two slant asymptotes: \(y=x+2\) and \(y=-x-2 .\) Use this fact to help sketch the curve.

A steel pipe is being carried down a hallway 9 ft wide. At the end of the hall there is a right-angled turn into a narrower hallway 6 ft wide. What is the length of the longest pipe that can be carried horizontally around the corner?