6.E: Applications of the Derivative (Exercises)

These are homework exercises to accompany David Guichard's "General Calculus" Textmap. Complementary General calculus exercises can be found for other Textmaps and can be accessed here.

6.1: Optimization

Ex 6.1.1 Let \(f(x) = \cases{1 + 4 x -x^2 & for x\leq3 \cr (x+5)/2 &for \(x>3\) \cr}\).

Find the maximum value and minimum values of \(f(x)\) for \(x\) in \([0,4]\). Graph \(f(x)\) to check your answers. (answer)

Ex 6.1.2 Find the dimensions of the rectangle of largest area having fixed perimeter 100. (answer)

Ex 6.1.3 Find the dimensions of the rectangle of largest area having fixed perimeter P. (answer)

Ex 6.1.4 A box with square base and no top is to hold a volume 100. Find the dimensions of the box that requires the least material for the five sides. Also find the ratio of height to side of the base. (answer)

Ex 6.1.5 A box with square base is to hold a volume 200. The bottom and top are formed by folding in flaps from all four sides, so that the bottom and top consist of two layers of cardboard. Find the dimensions of the box that requires the least material. Also find the ratio of height to side of the base. (answer)

Ex 6.1.6 A box with square base and no top is to hold a volume \(V\). Find (in terms of \(V\)) the dimensions of the box that requires the least material for the five sides. Also find the ratio of height to side of the base. (This ratio will not involve \(V\).) (answer)

Ex 6.1.7 You have 100 feet of fence to make a rectangular play area alongside the wall of your house. The wall of the house bounds one side. What is the largest size possible (in square feet) for the play area? (answer)

Ex 6.1.8 You have \(l\) feet of fence to make a rectangular play area alongside the wall of your house. The wall of the house bounds one side. What is the largest size possible (in square feet) for the play area? (answer)

Ex 6.1.9 Marketing tells you that if you set the price of an item at $10 then you will be unable to sell it, but that you can sell 500 items for each dollar below $10 that you set the price. Suppose your fixed costs total $3000, and your marginal cost is $2 per item. What is the most profit you can make?(answer)

Ex 6.1.10 Find the area of the largest rectangle that fits inside a semicircle of radius \(10\) (one side of the rectangle is along the diameter of the semicircle). (answer)

Ex 6.1.11 Find the area of the largest rectangle that fits inside a semicircle of radius \(r\) (one side of the rectangle is along the diameter of the semicircle). (answer)

Ex 6.1.12 For a cylinder with surface area 50, including the top and the bottom, find the ratio of height to base radius that maximizes the volume. (answer)

Ex 6.1.13 For a cylinder with given surface area \(S\), including the top and the bottom, find the ratio of height to base radius that maximizes the volume. (answer)

Ex 6.1.14 You want to make cylindrical containers to hold 1 liter using the least amount of construction material. The side is made from a rectangular piece of material, and this can be done with no material wasted. However, the top and bottom are cut from squares of side \(2r\), so that \(2(2r)^2=8r^2\) of material is needed (rather than \(2\pi r^2\), which is the total area of the top and bottom). Find the dimensions of the container using the least amount of material, and also find the ratio of height to radius for this container. (answer)

Ex 6.1.15 You want to make cylindrical containers of a given volume \(V\) using the least amount of construction material. The side is made from a rectangular piece of material, and this can be done with no material wasted. However, the top and bottom are cut from squares of side \(2r\), so that \(2(2r)^2=8r^2\) of material is needed (rather than \(2\pi r^2\), which is the total area of the top and bottom). Find the optimal ratio of height to radius. (answer)

Ex 6.1.16 Given a right circular cone, you put an upside-down cone inside it so that its vertex is at the center of the base of the larger cone and its base is parallel to the base of the larger cone. If you choose the upside-down cone to have the largest possible volume, what fraction of the volume of the larger cone does it occupy? (Let \(H\) and \(R\) be the height and base radius of the larger cone, and let \(h\) and \(r\) be the height and base radius of the smaller cone. Hint: Use similar triangles to get an equation relating \(h\) and \(r\).) (answer)

Ex 6.1.17 In example 6.1.12, what happens if \(w\ge v\)$ (i.e., your speed on sand is at least your speed on the road)? (answer)

Ex 6.1.18 A container holding a fixed volume is being made in the shape of a cylinder with a hemispherical top. (The hemispherical top has the same radius as the cylinder.) Find the ratio of height to radius of the cylinder which minimizes the cost of the container if (a) the cost per unit area of the top is twice as great as the cost per unit area of the side, and the container is made with no bottom; (b) the same as in (a), except that the container is made with a circular bottom, for which the cost per unit area is 1.5 times the cost per unit area of the side. (answer)

Ex 6.1.19 A piece of cardboard is 1 meter by \(1/2\) meter. A square is to be cut from each corner and the sides folded up to make an open-top box. What are the dimensions of the box with maximum possible volume? (answer)

Ex 6.1.20 (a) A square piece of cardboard of side \(a\) is used to make an open-top box by cutting out a small square from each corner and bending up the sides. How large a square should be cut from each corner in order that the box have maximum volume? (b) What if the piece of cardboard used to make the box is a rectangle of sides \(a\) and \(b\)? (answer)

Ex 6.1.21 A window consists of a rectangular piece of clear glass with a semicircular piece of colored glass on top; the colored glass transmits only \(1/2\) as much light per unit area as the the clear glass. If the distance from top to bottom (across both the rectangle and the semicircle) is 2 meters and the window may be no more than 1.5 meters wide, find the dimensions of the rectangular portion of the window that lets through the most light. (answer)

Ex 6.1.22 A window consists of a rectangular piece of clear glass with a semicircular piece of colored glass on top. Suppose that the colored glass transmits only \(k\) times as much light per unit area as the clear glass (\(k\) is between \(0\) and \(1\)). If the distance from top to bottom (across both the rectangle and the semicircle) is a fixed distance \(H\), find (in terms of \(k\)) the ratio of vertical side to horizontal side of the rectangle for which the window lets through the most light. (answer)

Ex 6.1.23 You are designing a poster to contain a fixed amount \(A\) of printing (measured in square centimeters) and have margins of \(a\) centimeters at the top and bottom and \(b\) centimeters at the sides. Find the ratio of vertical dimension to horizontal dimension of the printed area on the poster if you want to minimize the amount of posterboard needed. (answer)

Ex 6.1.24 The strength of a rectangular beam is proportional to the product of its width \(w\) times the square of its depth \(d\). Find the dimensions of the strongest beam that can be cut from a cylindrical log of radius \(r\). (answer)

Ex 6.1.25 What fraction of the volume of a sphere is taken up by the largest cylinder that can be fit inside the sphere? (answer)

Ex 6.1.26 The U.S. post office will accept a box for shipment only if the sum of the length and girth (distance around) is at most 108 in. Find the dimensions of the largest acceptable box with square front and back. (answer)

Ex 6.1.27 Find the dimensions of the lightest cylindrical can containing 0.25 liter (=250 cm \({}^3\)) if the top and bottom are made of a material that is twice as heavy (per unit area) as the material used for the side. (answer)

Ex 6.1.28 A conical paper cup is to hold \(1/4\) of a liter. Find the height and radius of the cone which minimizes the amount of paper needed to make the cup. Use the formula \(\pi r\sqrt{r^2+h^2}\) for the area of the side of a cone. (answer)

Ex 6.1.29 A conical paper cup is to hold a fixed volume of water. Find the ratio of height to base radius of the cone which minimizes the amount of paper needed to make the cup. Use the formula \( \pi r\sqrt{r^2+h^2}\) for the area of the side of a cone, called thelateral area of the cone. (answer)

Ex 6.1.30 If you fit the cone with the largest possible surface area (lateral area plus area of base) into a sphere, what percent of the volume of the sphere is occupied by the cone? (answer)

Ex 6.1.31 Two electrical charges, one a positive charge A of magnitude \(a\) and the other a negative charge B of magnitude \(b\), are located a distance \(c\) apart. A positively charged particle \(P\) is situated on the line between A and B. Find where \(P\) should be put so that the pull away from \(A\) towards \(B\) is minimal. Here assume that the force from each charge is proportional to the strength of the source and inversely proportional to the square of the distance from the source. (answer)

Ex 6.1.32 Find the fraction of the area of a triangle that is occupied by the largest rectangle that can be drawn in the triangle (with one of its sides along a side of the triangle). Show that this fraction does not depend on the dimensions of the given triangle. (answer)

Ex 6.1.33 How are your answers to Problem 9 affected if the cost per item for the \(x\) items, instead of being simply $2, decreases below $2 in proportion to \(x\) (because of economy of scale and volume discounts) by 1 cent for each 25 items produced? (answer)

Ex 6.1.34 You are standing near the side of a large wading pool of uniform depth when you see a child in trouble. You can run at a speed \( v_1\) on land and at a slower speed \( v_2\) in the water. Your perpendicular distance from the side of the pool is \(a\), the child's perpendicular distance is \(b\), and the distance along the side of the pool between the closest point to you and the closest point to the child is \(c\) (see the figure below). Without stopping to do any calculus, you instinctively choose the quickest route (shown in the figure) and save the child. Our purpose is to derive a relation between the angle \( \theta_1\) your path makes with the perpendicular to the side of the pool when you're on land, and the angle \(\theta_2\) your path makes with the perpendicular when you're in the water. To do this, let \(x\) be the distance between the closest point to you at the side of the pool and the point where you enter the water. Write the total time you run (on land and in the water) in terms of \(x\) (and also the constants \(a,b,c,v_1,v_2\)). Then set the derivative equal to zero. The result, called "Snell's law'' or the "law of refraction,'' also governs the bending of light when it goes into water. (answer)

6.2: Related Rates

Ex 6.2.1A cylindrical tank standing upright (with one circular base on the ground) has radius 20 cm. How fast does the water level in the tank drop when the water is being drained at 25 cm${}^3$/sec? (answer)

Ex 6.2.2A cylindrical tank standing upright (with one circular base on the ground) has radius 1 meter. How fast does the water level in the tank drop when the water is being drained at 3 liters per second? (answer)

Ex 6.2.3A ladder 13 meters long rests on horizontal ground and leans against a vertical wall. The foot of the ladder is pulled away from the wall at the rate of 0.6 m/sec. How fast is the top sliding down the wall when the foot of the ladder is 5 m from the wall? (answer)

Ex 6.2.4A ladder 13 meters long rests on horizontal ground and leans against a vertical wall. The top of the ladder is being pulled up the wall at $0.1$ meters per second. How fast is the foot of the ladder approaching the wall when the foot of the ladder is 5 m from the wall? (answer)

Ex 6.2.5A rotating beacon is located 2 miles out in the water. Let $A$ be the point on the shore that is closest to the beacon. As the beacon rotates at 10 rev/min, the beam of light sweeps down the shore once each time it revolves. Assume that the shore is straight. How fast is the point where the beam hits the shore moving at an instant when the beam is lighting up a point 2 miles along the shore from the point $A$? (answer)

Ex 6.2.6A baseball diamond is a square 90 ft on a side. A player runs from first base to second base at 15 ft/sec. At what rate is the player's distance from third base decreasing when she is half way from first to second base? (answer)

Ex 6.2.7Sand is poured onto a surface at 15 cm${}^3$/sec, forming a conical pile whose base diameter is always equal to its altitude. How fast is the altitude of the pile increasing when the pile is 3 cm high? (answer)

Ex 6.2.8A boat is pulled in to a dock by a rope with one end attached to the front of the boat and the other end passing through a ring attached to the dock at a point 5 ft higher than the front of the boat. The rope is being pulled through the ring at the rate of 0.6 ft/sec. How fast is the boat approaching the dock when 13 ft of rope are out? (answer)

Ex 6.2.9A balloon is at a height of 50 meters, and is rising at the constant rate of 5 m/sec. A bicyclist passes beneath it, traveling in a straight line at the constant speed of 10 m/sec. How fast is the distance between the bicyclist and the balloon increasing 2 seconds later? (answer)

Ex 6.2.10A pyramid-shaped vat has square cross-section and stands on its tip. The dimensions at the top are 2 m $\times$ 2 m, and the depth is 5 m. If water is flowing into the vat at 3 m${}^3$/min, how fast is the water level rising when the depth of water (at the deepest point) is 4 m? Note: the volume of any "conical'' shape (including pyramids) is $(1/3)(\hbox{height})(\hbox{area of base})$. (answer)

Ex 6.2.11The sun is rising at the rate of $1/4$ deg/min, and appears to be climbing into the sky perpendicular to the horizon, as depicted in figure 6.2.5. How fast is the shadow of a 200 meter building shrinking at the moment when the shadow is 500 meters long? (answer)

Ex 6.2.12The sun is setting at the rate of $1/4$ deg/min, and appears to be dropping perpendicular to the horizon, as depicted in figure 6.2.5. How fast is the shadow of a 25 meter wall lengthening at the moment when the shadow is 50 meters long? (answer)

Figure 6.2.5. Sunrise or sunset.

Ex 6.2.13The trough shown in figure 6.2.6is constructed by fastening together three slabs of wood of dimensions 10 ft $\times$ 1 ft, and then attaching the construction to a wooden wall at each end. The angle $\theta$ was originally $\ds 30^\circ$, but because of poor construction the sides are collapsing. The trough is full of water. At what rate (in ft${}^3$/sec) is the water spilling out over the top of the trough if the sides have each fallen to an angle of $\ds 45^\circ$, and are collapsing at the rate of $\ds 1^\circ$ per second? (answer)

Figure 6.2.6. Trough.

Ex 6.2.14A woman 5 ft tall walks at the rate of 3.5 ft/sec away from a streetlight that is 12 ft above the ground. At what rate is the tip of her shadow moving? At what rate is her shadow lengthening? (answer)

Ex 6.2.15A man 1.8 meters tall walks at the rate of 1 meter per second toward a streetlight that is 4 meters above the ground. At what rate is the tip of his shadow moving? At what rate is his shadow shortening? (answer)

Ex 6.2.16A police helicopter is flying at 150 mph at a constant altitude of 0.5 mile above a straight road. The pilot uses radar to determine that an oncoming car is at a distance of exactly 1 mile from the helicopter, and that this distance is decreasing at 190 mph. Find the speed of the car. (answer)

Ex 6.2.17A police helicopter is flying at 200 kilometers per hour at a constant altitude of 1 km above a straight road. The pilot uses radar to determine that an oncoming car is at a distance of exactly 2 kilometers from the helicopter, and that this distance is decreasing at 250 kph. Find the speed of the car. (answer)

Ex 6.2.18A light shines from the top of a pole 20 m high. A ball is falling 10 meters from the pole, casting a shadow on a building 30 meters away, as shown in figure 6.2.7. When the ball is 25 meters from the ground it is falling at 6 meters per second. How fast is its shadow moving? (answer)

Figure 6.2.7. Falling ball.

Ex 6.2.19Do example 6.2.6 assuming that the angle between the two roads is 120${}^\circ$ instead of 90${}^\circ$ (that is, the "north--south'' road actually goes in a somewhat northwesterly direction from $P$). Recall the law of cosines: $\ds c^2=a^2+b^2-2ab\cos\theta$. (answer)

Ex 6.2.20Do example 6.2.6 assuming that car A is 300 meters north of $P$, car B is 400 meters east of $P$, both cars are going at constant speed toward $P$, and the two cars will collide in 10 seconds. (answer)

Ex 6.2.21Do example 6.2.6 assuming that 8 seconds ago car A started from rest at $P$ and has been picking up speed at the steady rate of 5 m/sec${}^2$, and 6 seconds after car A started car B passed $P$ moving east at constant speed 60 m/sec. (answer)

Ex 6.2.22Referring again to example 6.2.6, suppose that instead of car B an airplane is flying at speed $200$ km/hr to the east of $P$ at an altitude of 2 km, as depicted in figure 6.2.8. How fast is the distance between car and airplane changing? (answer)

Here is some Sage code that will generate a 3D version of the figure. The result will display only if your browser has a java plugin.

Figure 6.2.8. Car and airplane.

Ex 6.2.23Referring again to example 6.2.6, suppose that instead of car B an airplane is flying at speed $200$ km/hr to the east of $P$ at an altitude of 2 km, and that it is gaining altitude at 10 km/hr. How fast is the distance between car and airplane changing? (answer)

Ex 6.2.24A light shines from the top of a pole 20 m high. An object is dropped from the same height from a point 10 m away, so that its height at time $\ds t$ seconds is $\ds h(t)=20-9.8t^2/2$. How fast is the object's shadow moving on the ground one second later? (answer)

Ex 6.2.25 The two blades of a pair of scissors are fastened at the point $A$ as shown in figure 6.2.9. Let $a$ denote the distance from $A$ to the tip of the blade (the point $B$). Let $\beta$ denote the angle at the tip of the blade that is formed by the line $\ds \overline{AB}$ and the bottom edge of the blade, line $\ds \overline{BC}$, and let $\theta$ denote the angle between $\ds \overline{AB}$ and the horizontal. Suppose that a piece of paper is cut in such a way that the center of the scissors at $A$ is fixed, and the paper is also fixed. As the blades are closed (i.e., the angle $\theta$ in the diagram is decreased), the distance $x$ between $A$ and $C$ increases, cutting the paper.

a. Express $x$ in terms of $a$, $\theta$, and $\beta$.

b. Express $dx/dt$ in terms of $a$, $\theta$, $\beta$, and $d\theta/dt$.

c. Suppose that the distance $a$ is 20 cm, and the angle $\beta$ is $\ds 5^\circ$. Further suppose that $\theta$ is decreasing at 50 deg/sec. At the instant when $\ds \theta=30^\circ$, find the rate (in cm/sec) at which the paper is being cut. (answer)

Figure 6.2.9. Scissors.

6.3: Newton's Method

Ex 6.3.1 Approximate the fifth root of 7, using \( x_0=1.5\) as a first guess. Use Newton's method to find \( x_3\) as your approximation. (answer)

Ex 6.3.2 Use Newton's Method to approximate the cube root of 10 to two decimal places. (answer)

Ex 6.3.3 The function \( f(x)=x^3-3x^2-3x+6\) has a root between 3 and 4, because \(f(3)=-3\) and \(f(4)=10\). Approximate the root to two decimal places. (answer)

Ex 6.3.4 A rectangular piece of cardboard of dimensions \(8\times 17\) is used to make an open-top box by cutting out a small square of side \(x\) from each corner and bending up the sides. (See exercise 20 in 6.1.) If \(x=2\), then the volume of the box is \(2\cdot 4\cdot 13=104\). Use Newton's method to find a value of \(x\) for which the box has volume 100, accurate to 3 significant figures. (answer)

6.4: Linear Approximations

Ex 6.4.1 Let \( f(x) = x^4\). If \(a=1\) and \(dx= \Delta x =1/2\), what are \(\Delta y\) and \(dy\)? (answer)

Ex 6.4.2 Let \( f(x) = \sqrt{x}\). If \(a=1\) and \(dx= \Delta x =1/10\), what are \(\Delta y\) and \(dy\)? (answer)

Ex 6.4.3 Let \(f(x) = \sin (2x)\). If \(a=\pi\) and \(dx= \Delta x =\pi/100\), what are \(\Delta y\) and \(dy\)? (answer)

Ex 6.4.4 Use differentials to estimate the amount of paint needed to apply a coat of paint 0.02 cm thick to a sphere with diameter \(40\) meters. (Recall that the volume of a sphere of radius \(r\) is \(V =(4/3)\pi r^3\). Notice that you are given that \(dr=0.02\).) (answer)

Ex 6.4.5 Show in detail that the linear approximation of \(\sin x\) at \(x=0\) is \(L(x)=x\) and the linear approximation of \(\cos x\) at \(x=0\) is \(L(x)=1\).

6.5: The Mean Value Theorem

Ex 6.5.1Let $\ds f(x) = x^2$. Find a value $c\in (-1,2)$ so that $f'(c)$ equals the slope between the endpoints of $f(x)$ on $[-1,2]$. (answer)

Ex 6.5.2Verify that $f(x) = x/(x+2)$ satisfies the hypotheses of the Mean Value Theorem on the interval $[1,4]$ and then find all of the values, $c$, that satisfy the conclusion of the theorem. (answer)

Ex 6.5.3Verify that $f(x) = 3x/(x+7)$ satisfies the hypotheses of the Mean Value Theorem on the interval $[-2 , 6]$ and then find all of the values, $c$, that satisfy the conclusion of the theorem.

Ex 6.5.4Let $f(x) = \tan x $. Show that $f(\pi ) = f(2\pi)=0$ but there is no number $c\in (\pi,2\pi)$ such that $f'(c) =0$. Why does this not contradict Rolle's theorem?

Ex 6.5.5Let $\ds f(x) = (x-3)^{-2}$. Show that there is no value $c\in (1,4)$ such that $f'(c) = (f(4)-f(1))/(4-1)$. Why is this not a contradiction of the Mean Value Theorem?

Ex 6.5.6Describe all functions with derivative $\ds x^2+47x-5$. (answer)

Ex 6.5.7Describe all functions with derivative $\ds {1\over 1+x^2}$. (answer)

Ex 6.5.8Describe all functions with derivative $\ds x^3-{1\over x}$. (answer)

Ex 6.5.9Describe all functions with derivative $\sin(2x)$. (answer)

Ex 6.5.10Show that the equation $\ds 6x^4 -7x+1 =0$ does not have more than two distinct real roots.

Ex 6.5.11Let $f$ be differentiable on $\R$. Suppose that $f'(x) \neq 0$ for every $x$. Prove that $f$ has at most one real root.

Ex 6.5.12Prove that for all real $x$ and $y$ $|\cos x -\cos y | \leq |x-y|$. State and prove an analogous result involving sine.

Ex 6.5.13Show that $\ds \sqrt{1+x} \le 1 +(x/2)$ if $-1 < x < 1$.