6.E: Applications of the Derivative (Exercises)
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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
Find the maximum value and minimum values of
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
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
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
Ex 6.1.11 Find the area of the largest rectangle that fits inside a semicircle of radius
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
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
Ex 6.1.15 You want to make cylindrical containers of a given volume
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
Ex 6.1.17 In example 6.1.12, what happens if
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
Ex 6.1.20 (a) A square piece of cardboard of side
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
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
Ex 6.1.23 You are designing a poster to contain a fixed amount
Ex 6.1.24 The strength of a rectangular beam is proportional to the product of its width
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
Ex 6.1.28 A conical paper cup is to hold
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
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
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
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
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)
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)
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)
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)
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)
6.3: Newton's Method
Ex 6.3.1 Approximate the fifth root of 7, using
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
Ex 6.3.4 A rectangular piece of cardboard of dimensions
6.4: Linear Approximations
Ex 6.4.1 Let
Ex 6.4.2 Let
Ex 6.4.3 Let
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
Ex 6.4.5 Show in detail that the linear approximation of
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 $-1 < x < 1$.