If u ∙ v = 0 and u × v = 0, then u = 0 or v = 0.
> 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
> 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 yea
> 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, 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?
> Gravel is being dumped from a conveyor belt at a rate of 30 ft3/min, and its coarseness is such that it forms a pile in the shape of a cone whose base diameter and height are always equal. How fast is the height of the pile increasing when the pile is 10
> If f (x) + x2 [f (x)]3 = 10 and f (1) = 2, find f ‘(1).
> A swimming pool is 20 ft wide, 40 ft long, 3 ft deep at the shallow end, and 9 ft deep at its deepest point. A crosssection is shown in the figure. If the pool is being filled at a rate of 0.8 ft3/min, how fast is the water level rising when the depth at
> A water trough is 10 m long and a cross-section has the shape of an isosceles trapezoid that is 30 cm wide at the bottom, 80 cm wide at the top, and has height 50 cm. If the trough is being filled with water at the rate of 0.2 m3/min, how fast is the wat
> A trough is 10 ft long and its ends have the shape of isosceles triangles that are 3 ft across at the top and have a height of 1 ft. If the trough is being filled with water at a rate of 12 ft3/min, how fast is the water level rising when the water is 6
> Water is leaking out of an inverted conical tank at a rate of 10,000 cm3/min at the same time that water is being pumped into the tank at a constant rate. The tank has height 6 m and the diameter at the top is 4 m. If the water level is rising at a rate
> A particle moves along the curve y = 2 sins(πx/2). As the particle passes through the point (1/3 , 1), its x-coordinate increases at a rate of 10 cm/s. How fast is the distance from the particle to the origin changing at this instant?
> At noon, ship A is 100 km west of ship B. Ship A is sailing south at 35 km/h and ship B is sailing north at 25 km/h. How fast is the distance between the ships changing at 4:00 pm?
> A boat is pulled into a dock by a rope attached to the bow of the boat and passing through a pulley on the dock that is 1 m higher than the bow of the boat. If the rope is pulled in at a rate of 1 m/s, how fast is the boat approaching the dock when it is
> The altitude of a triangle is increasing at a rate of 1 cm/min while the area of the triangle is increasing at a rate of 2 cm2/min. At what rate is the base of the triangle changing when the altitude is 10 cm and the area is 100 cm2?
> A baseball diamond is a square with side 90 ft. A batter hits the ball and runs toward first base with a speed of 24 ft/s. (a) At what rate is his distance from second base decreasing when he is halfway to first base? (b) At what rate is his distance fro
> A man starts walking north at 4 ft/s from a point P. Five minutes later a woman starts walking south at 5 ft/s from a point 500 ft due east of P. At what rate are the people moving apart 15 min after the woman starts walking?
> Find dy/dx by implicit differentiation. Tan(x – y) = y / 1 + x2
> Find f ’ (x) and f ’’ (x). f (x) = (x3 + 1) ex
> A spotlight on the ground shines on a wall 12 m away. If a man 2 m tall walks from the spotlight toward the building at a speed of 1.6 m/s, how fast is the length of his shadow on the building decreasing when he is 4 m from the building?
> Two cars start moving from the same point. One travels south at 60 mi/h and the other travels west at 25 mi/h. At what rate is the distance between the cars increasing two hours later?
> (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. At noon, ship A is 150 km west of ship B. Ship A
> (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 street light is mounted at the top of a 15-ft-
> For what values of x does the graph of f have a horizontal tangent? f (x) = ex cos x
