In this exercise we estimate the rate at which the total personal income is rising in the Richmond-Petersburg, Virginia, metropolitan area. In 1999, the population of this area was 961,400, and the population was increasing at roughly 9200 people per year. The average annual income was $30,593 per capita, and this average was increasing at about $1400 per year (a little above the national average of about $1225 yearly). Use the Product Rule and these figures to estimate the rate at which total personal income was rising in the Richmond-Petersburg area in 1999. Explain the meaning of each term in the Product Rule.
> 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)
> Find an equation of the tangent line to the curve at the given point. y = 2x3 – x2 +2, (1,3)
> Find equations of the tangent line and normal line to the given curve at the specified point. y = 2xex, (0, 0)
> Differentiate the function. y = ex+1 +1
> Differentiate the function. f(x) = 2 40
> Breathing is cyclic and a full respiratory cycle from the beginning of inhalation to the end of exhalation takes about 5 s. The maximum rate of air flow into the lungs is about 0.5 L/s. This explains, in part, why the function f (t) = 1/2 sin(2πt/5) has
> Find an equation of the tangent line to the given curve at the specified point. Y = 1+x / 1+ ex , (0 , 1/2)
> Differentiate the function. G(q) = (1 + q-1)2
> Differentiate the function. k(r) = er + re
> Differentiate the function. j(x) = x24 + e24
> Differentiate the function. h(u) = Au3 + Bu2 + Cu
> Find constants A and B such that the function y = A sin x + B cos x satisfies the differential equation y’’ + y’ - 2y = sin x.
> Differentiate the function. S(R) = 4πR2
> (a) Sketch, by hand, the graph of the function f (x) − ex, paying particular attention to how the graph crosses the y-axis. What fact allows you to do this? (b) What types of functions are f (x) − ex and t (x) − xe ? Compare the differentiation formulas
> Find the limit. lim θ→0 cos θ−1 ) sin θ
> An object with weight W is dragged along a horizontal plane by a force acting along a rope attached to the object. If the rope makes an angle with the plane, then the magnitude of the force is F = µW / µsinθ + cosθ where µ is a constant called the co
> A bacteria population starts with 400 bacteria and grows at a rate of r(t) = (450.268)e1.12567t bacteria per hour. How many bacteria will there be after three hours?
> A ladder 10 ft long rests against a vertical wall. Let be the angle between the top of the ladder and the wall and let x be the distance from the bottom of the ladder to the wall. If the bottom of the ladder slides away from the wall, how fast does x c
> For any vectors u and v in V3, |u × v | = |u ||v |.
> For any vectors u and v in V3, |u ∙ v | = |u ||v |.
> For any vectors u and v in V3, |u + v | = |u | + |v |.
> Find an equation of the tangent line to the given curve at the specified point. Y = x2 -1 / x2 + x + 1
> If u and v are in V3, then |u ∙ v | < |u | |v |.
> If u ∙ v = 0 and u × v = 0, then u = 0 or v = 0.
> If u × v = 0, then u = 0 or v = 0.
> An oil storage tank ruptures at time t = 0 and oil leaks from the tank at a rate of r(t) = 100e-0.01t liters per minute. How much oil leaks out during the first hour?
> If u ∙ v = 0, then u = 0 or v = 0.
> In R3 the graph of y = x2 is a paraboloid.
> The set of points {(x, y, z) | x2 + y2 = 1} is a circle.
> A linear equation Ax + By + Cz + D = 0 represents a line in space.
> The vector 3, −1, 2 is parallel to the plane 6x - 2y + 4z = 1
> For any vectors u and v in V3, (u + v) × v = u × v.
> For any vectors u and v in V3, (u × v) ∙ u = 0.
> An elastic band is hung on a hook and a mass is hung on the lower end of the band. When the mass is pulled downward and then released, it vibrates vertically. The equation of motion is s = 2 cos t + 3 sin t, t ≥ 0, where s is measured in centimeters and
> For any vectors u, v, and w in V3, u × (v × w) = (u × v) × w
> For any vectors u, v, and w in V3, u ∙ (v × w) = (u × v) ∙ w
> For any vectors u, v, and w in V3, (u + v) × w = u × w + v × w
> For any vectors u and v in V3 and any scalar k, K (u × v) = (ku) × v
> For any vectors u and v in V3 and any scalar k, K (u ∙ v) = (ku) ∙ v
> For any vectors u and v in V3, u × v = v × u.
> For any vectors u and v in V3, u ∙ v = v ∙ u.
> (a) Draw the line y = 2t + 1 and use geometry to find the area under this line, above the t-axis, and between the vertical lines t = 1 and t = 3. / //
> A mass on a spring vibrates horizontally on a smooth level surface (see the figure). Its equation of motion is x(t) = 8 sin t, where t is in seconds and x in centimeters. (a) Find the velocity and acceleration at time t. (b) Find the position, velocity,
> Which of the following areas are equal? Why? y. yA y= 2xe" y = evt y = esin x sin 2x 2
> Two sides of a triangle have lengths 12 m and 15 m. The angle between them is increasing at a rate of 2º /min. How fast is the length of the third side increasing when the angle between the sides of fixed length is 60°?
> Brain weight B as a function of body weight W in fish has been modeled by the power function B = 0.007W2/3, where B and W are measured in grams. A model for body weight as a function of body length L (measured in centimeters) is W = 0.12L2.53. If, over 1
> If g(x) + x sin g(x) = x2, find g’(0).
> When air expands adiabatically (without gaining or losing heat), its pressure P and volume V are related by the equation PV1.4 = C, where C is a constant. Suppose that at a certain instant the volume is 400 cm3 and the pressure is 80 kPa and is decreasin
> Boyle’s Law states that when a sample of gas is compressed at a constant temperature, the pressure P and volume V satisfy the equation PV = C, where C is a constant. Suppose that at a certain instant the volume is 600 cm3, the pressure is 150 kPa, and th
> A faucet is filling a hemispherical basin of diameter 60 cm with water at a rate of 2 L/min. Find the rate at which the water is rising in the basin when it is half full. [Use the following facts: 1 L is 1000 cm3. The volume of the portion of a sphere wi
> If the minute hand of a clock has length r (in centimeters), find the rate at which it sweeps out area as a function of r.
> According to the model we used to solve Example 2, what happens as the top of the ladder approaches the ground? Is the model appropriate for small values of y? Data from Example 2: A ladder 10 ft long rests against a vertical wall. If the bottom of the
> The top of a ladder slides down a vertical wall at a rate of 0.15 m/s. At the moment when the bottom of the ladder is 3 m from the wall, it slides away from the wall at a rate of 0.2 m/s. How long is the ladder?
> How fast is the angle between the ladder and the ground changing in Example 2 when the bottom of the ladder is 6 ft from the wall? Data from Example 2: A ladder 10 ft long rests against a vertical wall. If the bottom of the ladder slides away from the w
> The sides of an equilateral triangle are increasing at a rate of 10 cm/min. At what rate is the area of the triangle increasing when the sides are 30 cm long?
> A kite 100 ft above the ground moves horizontally at a speed of 8 ft/s. At what rate is the angle between the string and the horizontal decreasing when 200 ft of string has been let out?
