Get step-by-step solutions for your textbook problems from www.math4u.us
Larson R. Calculus, 8th Edition. Boston, Houghton Mifflin, 2006
Horizontal Tangent Line Determine the point(s) in the interval (0, 2 ) at which the graph of
f(x) = 2cos x + sin2x has a horizontal tangent.
Doppler Effect The frequency F of a fire truck siren heard by a stationary observer is where ±v represents the velocity of the accelerating fire truck in meters per second (see figure). Find the rate of change of F with respect to v when
(a) the fire truck is approaching at a velocity of 30 meters per second (use -v).
(b) the fire truck is moving away at a velocity of 30 meters per second (use v).
Harmonic Motion The displacement from equilibrium of an object in harmonic motion on the end of a spring is
y = 1/3 cos 12t - 1/4 sin 12t
where y is measured in feet and t is the time in seconds. Determine the position and velocity of the object when t = /8.
Pendulum A 15-centimeter pendulum moves according to the equation = 0.2cos 8t, where is the angular displacement from the vertical in radians and t is the time in seconds. Determine the maximum angular displacement and the rate of change of when t = 3 seconds.
Wave Motion A buoy oscillates in simple harmonic motion y = A cos t as waves move past it. The buoy moves a total of 3.5 feet (vertically) from its low point to its high point. It returns to its high point every 10 seconds.
(a) Write an equation describing the motion of the buoy if it is at its high point at t = 0.
(b) Determine the velocity of the buoy as a function of t.
Circulatory System The speed S of blood that is r centimeters from the center of an artery is
S = C(R2 - r2) where C is a constant, R is the radius of the artery, and S is measured in centimeters per second. Suppose a drug is administered and the artery begins to dilate at a rate of dR/dt. At a constant distance r find the rate at which S changes with respect to t for C =1.76 105, R =1.2 10-2 and
dR/dt = 10 -5
Use implicit differentiation to find an equation of the tangent line to the ellipse x2/2 + y2/8 =1 at (1,2)
Find the points at which the graph of the equation has a vertical or horizontal tangent line.
25x2 + 16y2 + 200x - 160y + 400 = 0
Weather Map The weather map shows several isobars- curves that represent areas of constant air pressure. Three high pressures H and one low pressure L are shown on the map. Given that wind speed is greatest along the orthogonal trajectories of the isobars, use the map to determine the areas having high wind speed.
Slope Find all points on the circle x2 + y2 = 25 where the slope is ?.
Horizontal Tangent Determine the point(s) at which the graph of y4 = y2 - x2 has a horizontal tangent.
Tangent Lines Find equations of both tangent lines to the ellipse x2/4 + y2/9 =1 that passes through the point (4,0).
Normals to a Parabola The graph shows the normal lines from the point (2,0)to the graph of the parabola x = y2. How many normal lines are there from the point (xo, 0) to the graph of the parabola if (a) x0 = ?; (b) x0 = ?; and (c) x0 = 1? For what value of x0 are two of the normal lines perpendicular to each other?
Find the rate of change of the distance between the origin and a moving point on the graph of
y = x2 + 1 if dx/dt = 2centimeters per second.
Find the rate of change of the distance between the origin and a moving point on the graph of y = sin x if dx/dt = 2centimeters per second.
Area The radius r of a circle is increasing at a rate of 3 centimeters per minute. Find the rates of change of the area when (a) r = 6 centimeters and (b) r = 24 centimeters.
Area Let A be the area of a circle of radius r that is changing with respect to time. If dr/dt is constant, is dA/dt constant? Explain.
Area The included angle of the two sides of constant equal length s of an isosceles triangle is .
(a) Show that the area of the triangle is given by A = ? s2sin .
(b) If is increasing at the rate of ? radian per minute, find the rates of change of the area when
= /6 and (c) = /3. Explain why the rate of change of the area of the triangle is not constant even though d /dt is constant.
Volume A spherical balloon is inflated with gas at the rate of 800 cubic centimeters per minute. How fast is the radius of the balloon increasing at the instant the radius is (a) 30 centimeters and (b) 60 centimeters?
Volume All edges of a cube are expanding at a rate of 3 centimeters per second. How fast is the volume changing when each edge is (a) 1 centimeter and (b) 10 centimeters?
Surface Area The conditions are the same as in Exercise 20. Determine how fast the surface area is changing when each edge is (a) 1 centimeter and (b) 10 centimeters.
Volume The formula for the volume of a cone is V = 1/3 r2h. Find the rate of change of the volume if dr/dt is 2 inches per minute and h = 3r when (a) r = 6 inches and (b) r = 24 inches.
Volume At a sand and gravel plant, sand is falling off a conveyor and onto a conical pile at a rate of 10 cubic feet per minute. The diameter of the base of the cone is approximately three times the altitude. At what rate is the height of the pile changing when the pile is 15 feet high?
Depth A conical tank (with vertex down) is 10 feet across the top and 12 feet deep. If water is flowing into the tank at a rate of 10 cubic feet per minute, find the rate of change of the depth of the water when the water is 8 feet deep.
Depth A swimming pool is 12 meters long, 6 meters wide, 1 meter deep at the shallow end, and 3 meters deep at the deep end (see figure). Water is being pumped into the pool at ? cubic meter per minute, and there is 1 meter of water at the deep end.
(a) What percent of the pool is filled?
(b) At what rate is the water level rising?
Depth A trough is 12 feet long and 3 feet across the top (see figure). Its ends are isosceles triangles with altitudes of 3 feet.
(a) If water is being pumped into the trough at 2 cubic feet per minute, how fast is the water level rising when h is 1 foot deep?
(b) If the water is rising at a rate of 3/8 inch per minute when h = 2 determine the rate at which water is being pumped into the trough.
Moving Ladder A ladder 25 feet long is leaning against the wall of a house (see figure). The base of the ladder is pulled away from the wall at a rate of 2 feet per second.
(a) How fast is the top of the ladder moving down the wall when its base is 7 feet, 15 feet, and 24 feet from the wall?
(b) Consider the triangle formed by the side of the house, the ladder, and the ground. Find the rate at which the area of the triangle is changing when the base of the ladder is 7 feet from the wall.
(c) Find the rate at which the angle between the ladder and the wall of the house is changing when the base of the ladder is 7 feet from the wall.
Construction A construction worker pulls a five-meter plank up the side of a building under construction by means of a rope tied to one end of the plank (see figure). Assume the opposite end of the plank follows a path perpendicular to the wall of the building and the worker pulls the rope at a rate of 0.15 meter per second. How fast is the end of the plank sliding along the ground when it is 2.5 meters from the wall of the building?
Construction A winch at the top of a 12-meter building pulls a pipe of the same length to a vertical position, as shown in the figure. The winch pulls in rope at a rate of -0.2 meter per second. Find the rate of vertical change and the rate of horizontal change at the end of the pipe when y=6.
Boating A boat is pulled into a dock by means of a winch 12 feet above the deck of the boat (see figure).
(a) The winch pulls in rope at a rate of 4 feet per second. Determine the speed of the boat when there is 13 feet of rope out. What happens to the speed of the boat as it gets closer to the dock?
(b) Suppose the boat is moving at a constant rate of 4 feet per second. Determine the speed at which the winch pulls in rope when there is a total of 13 feet of rope out. What happens to the speed at which the winch pulls in rope as the boat gets closer to the dock?
Air Traffic Control An air traffic controller spots two planes at the same altitude converging on a point as they fly at right angles to each other (see figure). One plane is 150 miles from the point moving at 450 miles per hour. The other plane is 200 miles from the point moving at 600 miles per hour.
(a) At what rate is the distance between the planes decreasing?
(b) How much time does the air traffic controller have to get one of the planes on a different flight path?
Air Traffic Control An airplane is flying at an altitude of 5 miles and passes directly over a radar antenna (see figure on previous page). When the plane is 10 miles away (s=10), the radar detects that the distance s is changing at a rate of 240 miles per hour. What is the speed of the plane?
Sports A baseball diamond has the shape of a square with sides 90 feet long (see figure). A player running from second base to third base at a speed of 28 feet per second is 30 feet from third base. At what rate is the player's distance s from home plate changing?
Sports For the baseball diamond in Exercise 33, suppose the player is running from first to second at a speed of 28 feet per second. Find the rate at which the distance from home plate is changing when the player is 30 feet from second base.
Shadow Length A man 6 feet tall walks at a rate of 5 feet per second away from a light that is 15 feet above the ground (see figure). When he is 10 feet from the base of the light,
(a) at what rate is the tip of his shadow moving?
(b) at what rate is the length of his shadow changing?
Shadow Length Repeat Exercise 35 for a man 6 feet tall walking at a rate of 5 feet per second toward a light that is 20 feet above the ground (see figure).
Machine Design The endpoints of a movable rod of length 1 meter have coordinates (x,0) and (0,y) (see figure). The position of the end on the x-axis is x(t) = ?(sin( t/6) where t is the time in seconds.
(a) Find the time of one complete cycle of the rod.
(b) What is the lowest point reached by the end of the rod on the y-axis?
(c) Find the speed of the y-axis endpoint when the axis endpoint is (1/4,0).
Evaporation As a spherical raindrop falls, it reaches a layer of dry air and begins to evaporate at a rate that is proportional to its surface area (S=4 r2). Show that the radius of the raindrop decreases at a constant rate.
Electricity The combined electrical resistance R of R1 and R2 connected in parallel, is given by
1/R = 1/R1 + 1/R2 where R, R1 and R2 are measured in ohms. R1 and R2 are increasing at rates of 1 and 1.5 ohms per second, respectively. At what rate R is changing when R1 = 50 ohms and R2 = 50 ohms?
Adiabatic Expansion When a certain polyatomic gas undergoes adiabatic expansion, its pressure p and volume V satisfy the equation pV1.3=k, where k is a constant. Find the relationship between the related rates dp/dt and dV/dt.
Roadway Design Cars on a certain roadway travel on a circular arc of radius r. In order not to rely on friction alone to overcome the centrifugal force, the road is banked at an angle of magnitude from the horizontal (see figure). The banking angle must satisfy the equation rg tan = v0 where v is the velocity of the cars and g = 32 feet per second per second is the acceleration due to gravity. Find the relationship between the related rates dv/dt and d /dt.
Angle of Elevation A balloon rises at a rate of 3 meters per second from a point on the ground 30 meters from an observer. Find the rate of change of the angle of elevation of the balloon from the observer when the balloon is 30 meters above the ground.
Angle of Elevation A fish is reeled in at a rate of 1 foot per second from a point 10 feet above the water (see figure). At what rate is the angle between the line and the water changing when there is a total of 25 feet of line out?
Angle of Elevation An airplane flies at an altitude of 5 miles toward a point directly over an observer (see figure). The speed of the plane is 600 miles per hour. Find the rates at which the angle of elevation is changing when the angle is (a) = 30o (b) = 60o and (c) = 75o.
Find the acceleration of the top of the ladder described in Exercise 27 when the base of the ladder is 7 feet from the wall.
Find the acceleration of the boat in Exercise 30(a) when there is a total of 13 feet of rope out.
Moving Shadow A ball is dropped from a height of 20 meters, 12 meters away from the top of a 20-meter lamppost (see figure). The ball's shadow, caused by the light at the top of the lamppost, is moving along the level ground. How fast is the shadow moving 1 second after the ball is released?
Vertical Motion A ball is dropped from a height of 100 feet. One second later, another ball is dropped from a height of 75 feet. Which ball hits the ground first?
Vertical Motion To estimate the height of a building, a weight is dropped from the top of the building into a pool at ground level. How high is the building if the splash is seen 9.2 seconds after the weight is dropped?
Vertical Motion A bomb is dropped from an airplane at an altitude of 14,400 feet. How long will it take for the bomb to reach the ground? (Because of the motion of the plane, the fall will not be vertical, but the time will be the same as that for a vertical fall.) The plane is moving at 600 miles per hour. How far will the bomb move horizontally after it is released from the plane?
A man 6 feet tall walks at a rate of 5 feet per second toward a streetlight that is 30 feet high (see figure). The man's 3-foot-tall child follows at the same speed, but 10 feet behind the man. At times, the shadow behind the child is caused by the man, and at other times, by the child.
(a) Suppose the man is 90 feet from the streetlight. Show that the man's shadow extends beyond the child's shadow.
(b) Suppose the man is 60 feet from the streetlight. Show that the child's shadow extends beyond the man's shadow.
(c) Determine the distance d from the man to the streetlight at which the tips of the two shadows are exactly the same distance from the streetlight.
(d) Determine how fast the tip of the shadow is moving as a function of x, the distance between the man and the street light. Discuss the continuity of this shadow speed function.