Chapter 4

The illumination at a point is inversely proportional to the square of the distance of the point from the light source and directly proportional to the intensity of the light source. If two light sources are \(s\) feet apart and their intensities are \(I_{1}\) and \(I_{2}\), respectively, at what point between them will the sum of their illuminations be a minimum?

Sketch the graph of a function \(f\) that has the following properties: (a) \(f\) is everywhere continuous; (b) \(f(0)=0, f(1)=2\); (c) \(f\) is an even function; (d) \(f^{\prime}(x)>0\) for \(x>0\); (e) \(f^{\prime \prime}(x)>0\) for \(x>0\).

Determine where the graph of the given function is increasing, decreasing, concave up, and concave down. Then sketch the graph (see Example 4). $$ g(x)=\frac{\ln (x+1)}{x+1} $$

Identify the critical points and find the extreme values on the interval \([-1,5]\) for each function: (a) \(f(x)=\cos x+x \sin x+2\) (b) \(g(x)=|f(x)|\)

Manhattan Island is said to have been bought by Peter Minuit in 1626 for \(\$ 24\). Suppose that Minuit had instead put the \(\$ 24\) in the bank at \(6 \%\) interest compounded continuously. What would that \(\$ 24\) have been worth in 2000 ?

What constant acceleration will cause a car to increase its velocity from 45 to 60 miles per hour in 10 seconds?

Consider the equation \(x=2\left(x-x^{2}\right)=g(x)\). (a) Sketch the graph of \(y=x\) and \(y=g(x)\) using the same coordinate system, and thereby approximately locate the positive root of \(x=g(x)\). (b) Try solving the equation by the Fixed-Point Algorithm starting with \(x_{1}=0.7\). (c) Solve the equation algebraically.

A wire of length 100 centimeters is cut into two pieces; one is bent to form a square, and the other is bent to form an equilateral triangle. Where should the cut be made if (a) the sum of the two areas is to be a minimum; (b) a maximum? (Allow the possibility of no cut.)

Find, if possible, the (global) maximum and minimum values of the given function on the indicated interval. $$ H(x)=\left|x^{2}-1\right| \text { on }[-2,2] $$

Sketch the graph of a function \(f\) that has the following properties: (a) \(f\) is everywhere continuous; (b) \(f(2)=-3, f(6)=1\); (c) \(f^{\prime}(2)=0, f^{\prime}(x)>0\) for \(x \neq 2, f^{\prime}(6)=3\); (d) \(f^{\prime \prime}(6)=0, f^{\prime \prime}(x)>0\) for \(26\).

In Problems 29-34, sketch the graph of a continuous function fon \([0,6]\) that satisfies all the stated conditions. \(f(0)=1 ; f(6)=3\); increasing and concave down on \((0.6)\)

Sketch the graph of a function with the given properties. \(f\) is differentiable, has domain \([0,6]\), reaches a maximum of 6 (attained when \(x=3\) ) and a minimum of 0 (attained when \(x=0\) ). Additionally, \(x=5\) is a stationary point.

If Methuselah's parents had put \(\$ 100\) in the bank for him at birth and he left it there, what would Methuselah have had at his death (969 years later) if interest was \(4 \%\) compounded annually?

A block slides down an inclined plane with a constant acceleration of 8 feet per second per second. If the inclined plane is 75 feet long and the block reaches the bottom in \(3.75\) seconds, what was the initial velocity of the block?

A closed box in the form of a rectangular parallelepiped with a square base is to have a given volume. If the material used in the bottom costs \(20 \%\) more per square inch than the material in the sides, and the material in the top costs \(50 \%\) more per square inch than that of the sides, find the most economical proportions for the box.

Find, if possible, the (global) maximum and minimum values of the given function on the indicated interval. $$ h(t)=\sin t^{2} \text { on }[0, \pi] $$

Sketch the graph of a function \(g\) that has the following properties: (a) \(g\) is everywhere smooth (continuous with a continuous first derivative); (b) \(g(0)=0\); (c) \(g^{\prime}(x)<0\) for all \(x\); (d) \(g^{\prime \prime}(x)<0\) for \(x<0\) and \(g^{\prime \prime}(x)>0\) for \(x>0\).

Sketch the graph of a continuous function fon \([0,6]\) that satisfies all the stated conditions. \(f(0)=8 ; f(6)=-2\); decreasing on \((0,6)\); inflection point at the ordered pair \((2,3)\), concave up on \((2,6)\)

Sketch the graph of a function with the given properties. \(f\) is differentiable, has domain \([0,6]\), reaches a maximum of 4 (attained when \(x=6\) ) and a minimum of \(-2\) (attained when \(x=1\) ). Additionally, \(x=2,3,4,5\) are stationary points.

Find the value of \(\$ 1000\) at the end of 1 year when the interest is compounded continuously at \(5 \%\). This is called the future value.

A certain rocket, initially at rest, is shot straight up with an acceleration of \(6 t\) meters per second per second during the first 10 seconds after blast- off, after which the engine cuts out and the rocket is subject only to gravitational acceleration of \(-10 \mathrm{me}\) 5ors per second per second. How high will the rocket go?

Consider \(x=\sqrt{1+x}\). (a) Apply the Fixed-Point Algorithm starting with \(x_{1}=0\) to find \(x_{2}, x_{3}, x_{4}\), and \(x_{5} .\) (b) Algebraically solve for \(x\) in \(x=\sqrt{1+x}\). (c) Evaluate \(\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}\).

Find, if possible, the (global) maximum and minimum values of the given function on the indicated interval. $$ f(x)=x e^{-x} \text { on }[0, \infty) $$

Sketch the graph of a function \(f\) that has the following properties: (a) \(f\) is everywhere continuous; (b) \(f(-3)=1\); (c) \(f^{\prime}(x)<0\) for \(x<-3, f^{\prime}(x)>0\) for \(x>-3, f^{\prime \prime}(x)<0\) for \(x \neq-3\).

Sketch the graph of a continuous function fon \([0,6]\) that satisfies all the stated conditions. $$ \begin{array}{l} f(0)=3 ; f(3)=0 ; f(6)=4 \\ f^{\prime}(x)<0 \text { on }(0,3) ; f^{\prime}(x)>0 \text { on }(3,6) ; \\ f^{\prime \prime}(x)>0 \text { on }(0,5) ; f^{\prime \prime}(x)<0 \text { on }(5,6) \end{array} $$

Sketch the graph of a function with the given properties. \(f\) is continuous, but not necessarily differentiable, has domain \([0,6]\), reaches a maximum of 6 (attained when \(x=5\) ), and a minimum of 2 (attained when \(x=3\) ). Additionally, \(x=1\) and \(x=5\) are the only stationary points.

Suppose that after 1 year you have \(\$ 1000\) in the bank. If the interest was compounded continuously at \(5 \%\), how much money did you put in the bank one year ago? This is called the present value.

Starting at station \(\mathrm{A}\), a commuter train accelerates at 3 meters per second per second for 8 seconds, then travels at constant speed \(v_{m}\) for 100 seconds, and finally brakes (decelerates) to a stop at station \(\mathrm{B}\) at 4 meters per second per second. Find (a) \(v_{m}\) and (b) the distance between \(\mathrm{A}\) and \(\mathrm{B}\).

Consider \(x=\sqrt{5+x}\). (a) Apply the Fixed-Point Algorithm starting with \(x_{1}=0\) to find \(x_{2}, x_{3}, x_{4}\), and \(x_{5} .\) (b) Algebraically solve for \(x\) in \(x=\sqrt{5+x}\). (c) Evaluate \(\sqrt{5+\sqrt{5+\sqrt{5+\cdots}}}\).

A weight connected to a spring moves along the \(x\) -axis so that its \(x\) -coordinate at time \(t\) is $$ x=\sin 2 t+\sqrt{3} \cos 2 t $$ What is the farthest that the weight gets from the origin?

Find, if possible, the (global) maximum and minimum values of the given function on the indicated interval. $$ g(x)=2^{x^{2}} \text { on }[-2,2] $$

Sketch the graph of a function \(f\) that has the following properties: (a) \(f\) is everywhere continuous; (b) \(f(-4)=-3, f(0)=0, f(3)=2\); (c) \(f^{\prime}(-4)=0, f^{\prime}(3)=0, f^{\prime}(x)>0\) for \(x<-4, f^{\prime}(x)>0\) for \(-43\) (d) \(f^{\prime \prime}(-4)=0, f^{\prime \prime}(0)=0, f^{\prime \prime}(x)<0\) for \(x<-4, f^{\prime \prime}(x)>0\) for \(-40\)

Sketch the graph of a continuous function fon \([0,6]\) that satisfies all the stated conditions. $$ \begin{array}{l} f(0)=3 ; f(2)=2 ; f(6)=0 \\ f^{\prime}(x)<0 \text { on }(0,2) \cup(2,6) ; f^{\prime}(2)=0 \\ f^{\prime \prime}(x)<0 \text { on }(0,1) \cup(2,6) ; f^{\prime \prime}(x)>0 \text { on }(1,2) \end{array} $$

Sketch the graph of a function with the given properties. \(f\) is continuous, but not necessarily differentiable, has domain \([0,6]\), reaches a maximum of 4 (attained when \(x=4)\), and a minimum of 2 (attained when \(x=2\) ). Additionally, \(f\) has no stationary points.

It will be shown later for small \(x\) that \(\ln (1+x) \approx x .\) Use this fact to show that the doubling time for money invested at \(p\) percent compounded annually is about \(70 / p\) years.

Starting from rest, a bus increases speed at constant acceleration \(a_{1}\), then travels at constant speed \(v_{m}\), and finally brakes to a stop at constant acceleration \(a_{2}\left(a_{2}<0\right) .\) It took 4 minutes to travel the 2 miles between stop \(\mathrm{C}\) and stop \(\mathrm{D}\) and then 3 minutes to go the \(1.4\) miles between stop \(\mathrm{D}\) and stop \(\mathrm{E}\). (a) Sketch the graph of the velocity \(v\) as a function of time \(t\), \(0 \leq t \leq 7\) (b) Find the maximum speed \(v_{m-}\) (c) If \(a_{1}=-a_{2}=a\), evaluate \(a\).

Prove: Let \(f\) be continuous on \([a, b]\) and differentiable on \((a, b)\). If \(f(a)\) and \(f(b)\) have opposite signs and if \(f^{\prime}(x) \neq 0\) for all \(x\) in \((a, b)\), then the equation \(f(x)=0\) has one and only one solution between \(a\) and \(b .\) Hint: Use the Intermediate Value Theorem and Rolle's Theorem (Problem 22).

A flower bed will be in the shape of a sector of a circle (a pie-shaped region) of radius \(r\) and vertex angle \(\theta .\) Find \(r\) and \(\theta\) if its area is a constant \(A\) and the perimeter is a minimum.

The first derivative \(f^{\prime}\) is given. Find all values of \(x\) that make the function \(f(a)\) a local minimum and \((b)\) a local maximum. \(f^{\prime}(x)=x^{3}(1-x)^{2}\)

Sketch the graph of a function \(f\) that has the following properties. (a) has a continuous first derivative; (b) is decreasing and concave up for \(x<3\) (c) has an extremum at \((3,1)\); (d) is increasing and concave up for \(36\).

Sketch the graph of a continuous function fon \([0,6]\) that satisfies all the stated conditions. $$ \begin{array}{l} f(0)=f(4)=1 ; f(2)=2 ; f(6)=0 ; \\ f^{\prime}(x)>0 \text { on }(0,2) ; f^{\prime}(x)<0 \text { on }(2,4) \cup(4,6) ; \\ f^{\prime}(2)=f^{\prime}(4)=0 ; f^{\prime \prime}(x)>0 \text { on }(0,1) \cup(3,4) \\ f^{\prime \prime}(x)<0 \text { on }(1,3) \cup(4,6) \end{array} $$

Sketch the graph of a function with the given properties. \(f\) is differentiable, has domain \([0,6]\), reaches a maximum of 4 (attained at two different values of \(x\), neither of which is an end point), and a minimum of 1 (attained at three different values of \(x\), exactly one of which is an end point.)

The equation for logistic growth is $$ \frac{d y}{d t}=k y(L-y) $$ Show that this differential equation has the solution $$ y=\frac{L y_{0}}{y_{0}+\left(L-y_{0}\right) e^{-L k t}} $$ Hint \(: \frac{1}{y(L-y)}=\frac{1}{L y}+\frac{1}{L(L-y)} .\)

A hot-air balloon left the ground rising at 4 feet per second. Sixteen seconds later, Victoria threw a ball straight up to her friend Colleen in the balloon. At what speed did she throw the ball if it just made it to Colleen?

Consider the equation \(x=x-f(x) / f^{\prime}(x)\) and suppose that \(f^{\prime}(x) \neq 0\) in an interval \([a, b]\). (a) Show that if \(r\) is in \([a, b]\) then \(r\) is a root of the equation \(x=x-f(x) / f^{\prime}(x)\) if and only if \(f(r)=0\) (b) Show that Newton's Method is a special case of the FixedPoint Algorithm. in which \(g^{\prime}(r)=0\)

The first derivative \(f^{\prime}\) is given. Find all values of \(x\) that make the function \(f(a)\) a local minimum and \((b)\) a local maximum. \(f^{\prime}(x)=-(x-1)(x-2)(x-3)(x-4)\)

Linear approximations provide particularly good approximations near points of inflection. Using a graphing calculator, investigate this behavior in Problems 34-36. Graph \(y=\sin x\) and its linear approximation \(L(x)=x\) at \(x=0\).

Sketch the graph of a continuous function fon \([0,6]\) that satisfies all the stated conditions. $$ \begin{array}{l} f(0)=f(3)=3 ; f(2)=4 ; f(4)=2 ; f(6)=0 \\ f^{\prime}(x)>0 \text { on }(0,2) ; f^{\prime}(x)<0 \text { on }(2,4) \cup(4,5) ; \\ f^{\prime}(2)=f^{\prime}(4)=0 ; f^{\prime}(x)=-1 \text { on }(5,6) \\ f^{\prime \prime}(x)<0 \text { on }(0,3) \cup(4,5) ; f^{\prime \prime}(x)>0 \text { on }(3,4) \end{array} $$

Sketch the graph of a function with the given properties. \(f\) is continuous but not necessarily differentiable, has domain \([0,6]\), reaches a maximum of 6 (attained when \(x=0\) ) and a minimum of \(0(\) attained when \(x=6\) ). Additionally, \(f\) has two stationary points and two singular points in \((0,6)\).

According to Torricelli's Law, the time rate of change of the volume \(V\) of water in a draining tank is proportional to the square root of the water's depth. A cylindrical tank of radius \(10 / \sqrt{\pi}\) centimeters and height 16 centimeters, which was full initially, took 40 seconds to drain. (a) Write the differential equation for \(V\) at time \(t\) and the two corresponding conditions. (b) Solve the differential equation. (c) Find the volume of water after 10 seconds.