Show that if a ≠ 0 and b ≠ 0, then there exist numbers α and β such that aex + be-x equals either α sinh(x + β) or α cosh(x + β) In other words, almost every function of the form f (x) = aex + be-x is a shifted and stretched hyperbolic sine or cosine function.
> If a current I passes through a resistor with resistance R, Ohm’s Law states that the voltage drop is V = RI. If V is constant and R is measured with a certain error, use differentials to show that the relative error in calculating I is approximately the
> A manufacturer produces bolts of a fabric with a fixed width. The quantity q of this fabric (measured in yards) that is sold is a function of the selling price p (in dollars per yard), so we can write q − f (p). Then the total revenue earned with selling
> One side of a right triangle is known to be 20 cm long and the opposite angle is measured as 30°, with a possible error of 61°. (a) Use differentials to estimate the error in computing the length of the hypotenuse. (b) What is the percentage error?
> (a) Use differentials to find a formula for the approximate volume of a thin cylindrical shell with height h, inner radius r, and thickness Dr. (b) What is the error involved in using the formula from part (a)?
> Use differentials to estimate the amount of paint needed to apply a coat of paint 0.05 cm thick to a hemispherical dome with diameter 50 m.
> A plane flies horizontally at an altitude of 5 km and passes directly over a tracking telescope on the ground. When the angle of elevation is π/3, this angle is decreasing at a rate of π/6 rad/min. How fast is the plane traveling at that time?
> The circumference of a sphere was measured to be 84 cm with a possible error of 0.5 cm. (a) Use differentials to estimate the maximum error in the calculated surface area. What is the relative error? (b) Use differentials to estimate the maximum error i
> The radius of a circular disk is given as 24 cm with a maximum error in measurement of 0.2 cm. (a) Use differentials to estimate the maximum error in the calculated area of the disk. (b) What is the relative error? What is the percentage error?
> The edge of a cube was found to be 30 cm with a possible error in measurement of 0.1 cm. Use differentials to estimate the maximum possible error, relative error, and percentage error in computing (a) the volume of the cube and (b) the surface area of th
> Let f (x) = (x – 1)2 g(x) = e-2x and h(x) = 1 + ln(1 - 2x) (a) Find the linearizations of f, t, and h at a = 0. What do you notice? How do you explain what happened? (b) Graph f, t, and h and their linear approximations. For which function is the linear
> Explain, in terms of linear approximations or differentials, why the approximation is reasonable. 1 9.98 ≈ 0.1002
> Explain, in terms of linear approximations or differentials, why the approximation is reasonable. 4.02 ≈ 2.005
> Explain, in terms of linear approximations or differentials, why the approximation is reasonable. sec 0.08 ≈ 1
> Use implicit differentiation to find an equation of the tangent line to the curve at the given point. Sin(x + y) = 2x - 2y, (π , π )
> Use a linear approximation (or differentials) to estimate the given number. cos 29°
> Use a linear approximation (or differentials) to estimate the given number. e0.1
> A lighthouse is located on a small island 3 km away from the nearest point P on a straight shoreline and its light makes four revolutions per minute. How fast is the beam of light moving along the shoreline when it is 1 km from P?
> Use a linear approximation (or differentials) to estimate the given number. 100.5
> Use a linear approximation (or differentials) to estimate the given number. 3 1001
> Use a linear approximation (or differentials) to estimate the given number. 1/4.002
> Use a linear approximation (or differentials) to estimate the given number. (1.999)4
> Compute ∆y and dy for the given values of x and dx = ∆x. Then sketch a diagram like Figure 5 showing the line segments with lengths dx, dy, and ∆y. y = ex, x = 0, ∆x = 0.5 Figure 5:
> Compute ∆y and dy for the given values of x and dx = ∆x. Then sketch a diagram like Figure 5 showing the line segments with lengths dx, dy, and ∆y. y = x - x3, x = 0, ∆x = 20.3 Figu
> Compute ∆y and dy for the given values of x and dx = ∆x. Then sketch a diagram like Figure 5 showing the line segments with lengths dx, dy, and ∆y. y = x2 - 4x, x = 3, ∆x = 0.5 Figu
> Use implicit differentiation to find an equation of the tangent line to the curve at the given point. y sin 2x = x cos 2y, ( π/2, π/ 4)
> A television camera is positioned 4000 ft from the base of a rocket launching pad. The angle of elevation of the camera has to change at the correct rate in order to keep the rocket in sight. Also, the mechanism for focusing the camera has to take into a
> (a) Find the differential dy and (b) evaluate dy for the given values of x and dx. y = cos x, x = 13, dx = 20.02
> (a) Find the differential dy and (b) evaluate dy for the given values of x and dx. y = ex/10, x = 0, dx = 0.1
> Verify the given linear approximation at a = 0. Then determine the values of x for which the linear approximation is accurate to within 0.1. ex cos x ≈ 1 + x
> Regard y as the independent variable and x as the dependent variable and use implicit differentiation to find dx/dy. y sec x = x tan y
> Two carts, A and B, are connected by a rope 39 ft long that passes over a pulley P (see the figure). The point Q is on the floor 12 ft directly beneath P and between the carts. Cart A is being pulled away from Q at a speed of 2 ft/s. How fast is cart B m
> Verify the given linear approximation at a = 0. Then determine the values of x for which the linear approximation is accurate to within 0.1. ln (1 + x) ≈ x
> The length of a rectangle is increasing at a rate of 8 cm/s and its width is increasing at a rate of 3 cm/s. When the length is 20 cm and the width is 10 cm, how fast is the area of the rectangle increasing?
> The tangent line approximation L(x) is the best first-degree (linear) approximation to f (x) near x = a because f (x) and L(x) have the same rate of change (derivative) at a. For a better approximation than a linear one, let’s try a second-degree (quadra
> Each side of a square is increasing at a rate of 6 cm/s. At what rate is the area of the square increasing when the area of the square is 16 cm2?
> The tangent line approximation L(x) is the best first-degree (linear) approximation to f (x) near x = a because f (x) and L(x) have the same rate of change (derivative) at a. For a better approximation than a linear one, let’s try a second-degree (quadra
> The tangent line approximation L(x) is the best first-degree (linear) approximation to f (x) near x = a because f (x) and L(x) have the same rate of change (derivative) at a. For a better approximation than a linear one, let’s try a second-degree (quadra
> The tangent line approximation L(x) is the best first-degree (linear) approximation to f (x) near x = a because f (x) and L(x) have the same rate of change (derivative) at a. For a better approximation than a linear one, let’s try a second-degree (quadra
> 1. Consider the family of curves y2 - 2x2(x + 8) = [(y + 1)2(y + 9) - x2] (a) By graphing the curves with c = 0 and c = 2, determine how many points of intersection there are. (You might have to zoom in to find all of them.) (b) Now add the curves with c
> (a) Graph several members of the family of curves x2 + y2 + cx2y2 = 1 Describe how the graph changes as you change the value of c. (b) What happens to the curve when c = -1? Describe what appears on the screen. Can you prove it algebraically? (c) Find y’
> An approach path for an aircraft landing is shown in the figure and satisfies the following conditions: (i) The cruising altitude is h when descent starts at a horizontal distance, from touchdown at the origin. (ii) The pilot must maintain a constant hor
> (a) The curve with equation y2 = 5x4 - x2 is called a kampyle of Eudoxus. Find an equation of the tangent line to this curve at the point (1, 2). (b) Illustrate part (a) by graphing the curve and the tangent line on a common screen. (If your graphing dev
> Suppose you are asked to design the first ascent and drop for a new roller coaster. By studying photographs of your favorite coasters, you decide to make the slope of the ascent 0.8 and the slope of the drop -1.6. You decide to connect these two straight
> (a) If A is the area of a circle with radius r and the circle expands as time passes, find dA/dt in terms of dr/dt. (b) Suppose oil spills from a ruptured tanker and spreads in a circular pattern. If the radius of the oil spill increases at a constant ra
> Find the numerical value of each expression. (a) sech 0 (b) cosh-1 1
> Find the numerical value of each expression. (a) sinh 4 (b) sinh(ln 4)
> Find the numerical value of each expression. (a) cosh(ln 5) (b) cosh 5
> Find the numerical value of each expression. (a) tanh 0 (b) tanh 1
> Find the numerical value of each expression. (a) sinh 0 (b) cosh 0
> Investigate the family of functions fn(x) = tanh (n sin x) where n is a positive integer. Describe what happens to the graph of fn when n becomes large.
> At what point of the curve y = cosh x does the tangent have slope 1?
> If x = ln(sec θ + tan θ), show that sec θ = cosh x.
> Use implicit differentiation to find an equation of the tangent line to the curve at the given point. y2 (y2 – 4) = x2(x2 – 5), (0, -2), (devil’s curve) y
> If V is the volume of a cube with edge length x and the cube expands as time passes, find dV/dt in terms of dx/dt.
> (a) Show that any function of the form y = A sinh mx + B cosh mx satisfies the differential equation y’’ = m2y. (b) Find y = y(x) such that y’’ = 9y, y(0) = -4, and y’(0) = 6.
> A telephone line hangs between two poles 14 m apart in the shape of the catenary y = 20 cosh(x/20) - 15, where x and y are measured in meters. (a) Find the slope of this curve where it meets the right pole. (b) Find the angle θ between the l
> The Gateway Arch in St. Louis was designed by Eero Saarinen and was constructed using the equation y = 211.49 - 20.96 cosh 0.03291765x for the central curve of the arch, where x and y are measured in meters and |x | ≤ 91.20. (a) Graph the central curve.
> The minute hand on a watch is 8 mm long and the hour hand is 4 mm long. How fast is the distance between the tips of the hands changing at one o’clock?
> Use implicit differentiation to find an equation of the tangent line to the curve at the given point. 2(x2 + y2)2 = 25(x2 - y2), (3, 1), (lemniscate) yA
> Find the derivative. Simplify where possible. y = coth-1 (sec x)
> Find the derivative. Simplify where possible. y = sech-1 (e-x)
> Find the derivative. Simplify where possible. y = x sinh-1 (x/3)
> Find the derivative. Simplify where possible. y = x tanh-1 x
> Find the derivative. Simplify where possible. Yy = sinh-1 (tanx)
> Find the derivative. Simplify where possible. y = e cosh 3x
> A runner sprints around a circular track of radius 100 m at a constant speed of 7 m/s. The runner’s friend is standing at a distance 200 m from the center of the track. How fast is the distance between the friends changing when the distance between them
> (a) What quantities are given in the problem? (b) What is the unknown? (c) Draw a picture of the situation for any time t. (d) Write an equation that relates the quantities. (e) Finish solving the problem. If a snowball melts so that its surface area dec
> (a) What quantities are given in the problem? (b) What is the unknown? (c) Draw a picture of the situation for any time t. (d) Write an equation that relates the quantities. (e) Finish solving the problem. A plane flying horizontally at an altitude of 1
> A particle is moving along a hyperbola xy = 8. As it reaches the point (4, 2), the y-coordinate is decreasing at a rate of 3 cm/s. How fast is the x-coordinate of the point changing at that instant?
> If x2 + y2 + z2 = 9, dx/dt = 5, and dy/dt = 4, find dz/dt when (x, y, z) = (2, 2, 1).
> Suppose 4x2 + 9y2 = 36, where x and y are functions of t. (a) If dy/dt = 1/3, find dx/dt when x = 2 and y = 2 3 5 . (b) If dx/dt = 3, find dy/dt when x = -2 and y = 2 3 5 .
> Find dy/dx by implicit differentiation. Sin(xy) = cos(x + y)
> The area of a triangle with sides of lengths a and b and contained angle is θ A = 1/2 ab sin θ (a) If a = 2 cm, b = 3 cm, and increases at a rate of 0.2 rad/min, how fast is the area increasing when θ = π / 3? (b) If a = 2 cm, b increases at a rate
> The radius of a spherical ball is increasing at a rate of 2 cm/min. At what rate is the surface area of the ball increasing when the radius is 8 cm?
> The radius of a sphere is increasing at a rate of 4 mm/s. How fast is the volume increasing when the diameter is 80 mm?
> A cylindrical tank with radius 5 m is being filled with water at a rate of 3 m3/min. How fast is the height of the water increasing?
> Strontium-90 has a half-life of 28 days. (a) A sample has a mass of 50 mg initially. Find a formula for the mass remaining after t days. (b) Find the mass remaining after 40 days. (c) How long does it take the sample to decay to a mass of 2 mg? (d) Sketc
> Experiments show that if the chemical reaction N2O5 ( 2NO2 + ½ O2 takes place at 458C, the rate of reaction of dinitrogen pentoxide is proportional to its concentration as follows: d[N2O5] / dt = 0.0005[N2O5] (See Example 3.7.4.) (a) Find an expression
> The table gives the population of Indonesia, in millions, for the second half of the 20th century. (a) Assuming the population grows at a rate proportional to its size, use the census figures for 1950 and 1960 to predict the population in 1980. Compare
> The table gives estimates of the world population, in millions, from 1750 to 2000. (a) Use the exponential model and the population figures for 1750 and 1800 to predict the world population in 1900 and 1950. Compare with the actual figures. (b) Use the
> A bacteria culture grows with constant relative growth rate. The bacteria count was 400 after 2 hours and 25,600 after 6 hours. (a) What is the relative growth rate? Express your answer as a percentage. (b) What was the intitial size of the culture? (c)
> Find the first and second derivatives of the function. Check to see that your answers are reasonable by comparing the graphs of f, f ‘, and f ‘’. f(x) = ex – x3
> Find dy/dx by implicit differentiation. x sin y + y sin x = 1
> Differentiate the function. f(x) = 5.2x + 2.3
> Find the first and second derivatives of the function. Check to see that your answers are reasonable by comparing the graphs of f, f ‘, and f ‘’. f(x) = 2x – 5x3/4
> Find the absolute maximum and absolute minimum values of f on the given interval. f (x) = 5 + 54x - 2x3, [0, 4]
> Find the first and second derivatives of the function. f(x) = 0.001 x5 – 0.02 x3
> (a) Graph the function g(x) − ex - 3x2 in the viewing rectangle [-1, 4] by [-8, 8]. (b) Using the graph in part (a) to estimate slopes, make a rough sketch, by hand, of the graph of t9. (See Example 2.8.1.) (c) Calculate t9sxd and use this expression, wi
> (a) Graph the function f(x) = x4 – 3x3 – 6x2 + 7x + 30 in the viewing rectangle [-3,5] by [-10,50]. (b) Using the graph in part (a) to estimate slopes, make a rough sketch, by hand, of the graph of f’. (See Example 2.8.1.) (c) Calculate f’(x) and use th
> Find f’(x). Compare the graphs of f and f’ and use them to explain why your answer is reasonable. f(x) — х5 — — 2x3 + x 1
> Find f’(x). Compare the graphs of f and f’ and use them to explain why your answer is reasonable. f(x) = x4 – 2x3 + x2
> Find an equation of the tangent line to the curve at the given point. Illustrate by graphing the curve and the tangent line on the same screen. y = x - Vx, (1, 0)
> Find equations of the tangent line and normal line to the given curve at the specified point. y = 2x / x2 + 1 , (1, 1)
> Find dy/dx by implicit differentiation. tan-1(x2y) = x + xy2
> Find an equation of the tangent line to the curve at the given point. Illustrate by graphing the curve and the tangent line on the same screen. y = 3x2 – x3, (1, 2)
> Differentiate the function. f(x) = e5
> Find equations of the tangent line and normal line to the curve at the given point. y2 = x3, (1,1)
> Find equations of the tangent line and normal line to the curve at the given point. y= x4 + 2ex, (0, 2)
> Find an equation of the tangent line to the curve at the given point. y = x + 2/x, (2, 3)
> Find an equation of the tangent line to the curve at the given point. y = 2ex + x, (0, 2)
