Question: An observer stands at a point P,
An observer stands at a point P, one unit away from a track. Two runners start at the point S in the figure and run along the track. One runner runs three times as fast as the other. Find the maximum value of the observer’s angle of sight θ between the runners. [Hint: Maximize tan θ.]
![An observer stands at a point P, one unit away from a track. Two runners start at the point S in the figure and run along the track. One runner runs three times as fast as the other. Find the maximum value of the observer’s angle of sight θ between the runners. [Hint: Maximize tan θ.]](../images/an-observer-stands-at-a-point-p-one-unit-away-from-188466-1.jpg)
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> Evaluate the integral. f x/√1 – x4, dx
> Evaluate the integral. f12 (x – 1)3/x2 dx
> Evaluate f10 (x + √1 – x2) dx by interpreting it in terms of areas.
> Evaluate the integral. f41 x3/2 lnx dx
> Evaluate the integral. f41 dt/ (2t + 1)3
> Evaluate the integral. fπ/4-π/4 t4 tan t/2 + cos t, dt
> Evaluate the integral. f50 ye-0.6y dy
> Evaluate the integral. f50 x/x + 10, dx
> Evaluate the integral. f x + 2/√x2 + 4x, dx
> Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. x² – y? = a², x = a + h (where a > 0, h> 0); about the y-axis
> Evaluate the integral. f21 ½ - 3x, dx
> A stone is dropped from the upper observation deck (the Space Deck) of the CN Tower, 450m above the ground. (a). Find the distance of the stone above ground level at time t. (b). How long does it take the stone to reach the ground? (c). With what velocit
> Evaluate the integral. f0π/4 1 + cos2θ/cos2 θ dθ
> (a). Evaluate the Riemann sum for f (x) = x2 – x, 0 (b). Use the definition of a definite integral (with right end - points) to calculate the value of the integral (c). Use the Evaluation Theorem to check your answer to part (b). (d).
> Evaluate the integral. f10 eπt dt
> Evaluate the integral. f10 sin (3πt) dt
> Evaluate the integral. f10v2 cos (v3) dv
> Evaluate the integral. f csc2x/ 1 + cot x, dx
> Evaluate the integral. f10 x/x2 + 1, dx
> Evaluate the integral. f10 (4√u + 1)2 du
> Evaluate the integral. f (1 – x/x)2 dx
> Evaluate the integral. f10 (1 – x)9 dx
> Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. y = x' + 1, y= 9 – x²; about y = -1
> Evaluate the integral. f0π/3 sin θ + sin θ tan2 θ /sec2θ, dθ
> (a). If g (x) = 1/(√x -1), use your calculator or computer to make a table of approximate values of f12g (x) dx for t = 5, 10, 100, 1000, and 10,000. Does it appear that f∞2g (x) dx is convergent or divergent? (b). Use the Comparison Theorem with f (x) =
> Evaluate the integral. f10 (1 – x9) dx
> Evaluate the integral. fr0 (x4 – 8x + 7) dx
> Use the given graph of f to find the Riemann sum with six subintervals. Take the sample points to be (a) left endpoints and (b) midpoints. In each case draw a diagram and explain what the Riemann sum represents. y= f(x) 6 2. 2.
> (a). Find the vertical and horizontal asymptotes, if any. (b). Find the intervals of increase or decrease. (c). Find the local maximum and minimum values. (d). Find the intervals of concavity and the inflection points. (e). Use the information from parts
> (a). Find the vertical and horizontal asymptotes, if any. (b). Find the intervals of increase or decrease. (c). Find the local maximum and minimum values. (d). Find the intervals of concavity and the inflection points. (e). Use the information from parts
> A light is to be placed atop a pole of height feet to illuminate a busy traffic circle, which has a radius of 40 ft. The intensity of illumination I at any point P on the circle is directly proportional to the cosine of the angle (see the figure) and inv
> A canister is dropped from a helicopter 500 m above the ground. Its parachute does not open, but the canister has been designed to withstand an impact velocity of 100 m/s. Will it burst?
> Sketch the graph of a continuous, even function f such that f (0) =, f'(x) = 2x if 0 < x < 1, f'(x) = -1 if 1 < x < 3, and f'(x) = 1 if x > 3.
> Find the local and absolute extreme values of the function on the given interval. f(x) = (In x)/x², [1, 3]
> Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. 1. If f'(c) = 0, then f has a local maximum or minimum at c. 2. If f has an absolute minimum value
> A particle is moving with the given data. Find the position of the particle. a (t) = sin t + 3 cos t, s (0) = 0 v (0) = 2
> Show that if f is a polynomial of degree 3 or lower, then Simpson’s Rule gives the exact value of fba f (x) dx.
> Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. x = 0, x= 9 – y?; about x = -1
> A particle is moving with the given data. Find the position of the particle. v (t) = 2t – 1/ (1 + t2), s (0) = 1
> Find f (x). f"(x) = 2x3 + 3x2 - 4x + 5 f (0) = 2, f(1) = 0
> Find f (x). f"(x) = 1 - 6x + 48x2 f (0) = 1 f'(0) = 2
> Find f (x). f' (u) = u2 + √u /u, f (1) = 3
> Find f (x). f'(t) = 2t - 3 sin t, f (0) = 5
> Use Newton’s method to find all roots of the equation sin x = x2 – 3x + 1 correct to six decimal places.
> Find the local and absolute extreme values of the function on the given interval. f(x) = x + sin 2x, [0, 7]
> 1. If f and g are continuous on [a, b], then 2. If f and g are continuous on [a, b], then 3. If f is continuous on [a, b], then 4. If f is continuous on [a, b], then 5. If f is continuous on [a, b] and f (x) > 0, then 7. If f and g are cont
> A metal storage tank with volume V is to be constructed in the shape of a right circular cylinder surmounted by a hemisphere. What dimensions will require the least amount of metal?
> The velocity of a wave of length L in deep water is where K and C are known positive constants. What is the length of the wave that gives the minimum velocity? L v = K- C C L
> Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. x = 1+ y', y = x – 3; about the y-axis
> Find the area of the region bounded by the given curves. y = x², y= 4x – x²
> (a). What is the physical significance of the center of mass of a thin plate? (b). If the plate lies between y = f (x) and y = 0, where a < x < b, write expressions for the coordinates of the center of mass.
> If you have a graphing calculator or computer, why do you need calculus to graph a function?
> Suppose that you push a book across a 6-meter-long table by exerting a force f (x) at each point from x = 0 to x = 6. What does f60f (x) dx represent? If is measured in newtons, what are the units for the integral?
> (a). Explain the meaning of the indefinite integral ff (x) dx. (b). What is the connection between the definite integral fbaf (x)dx and the indefinite integral ff (x) dx?
> An arc PQ of a circle subtends a central angle θ as in the figure. Let A (θ) be the area between the chord PQ and the arc PQ. Let B (θ) be the area between the tangent lines PR, QR and the arc. Find A(0) lim 0
> (a). How is the length of a curve defined? (b). Write an expression for the length of a smooth curve with parametric equations x = f (t), y = g (t), a < t < b. (c). How does the expression in part (b) simplify if the curve is described by giving y in ter
> If r (t) is the rate at which water flows into a reservoir, what does ft2t1r (t) dt represent?
> (a). Write the definition of the definite integral of a continuous function from a to b. (b). What is the geometric interpretation of fba f (x) dx if f (x) > 0? (c). What is the geometric interpretation of fba f (x) dx if f (x) takes on both positive and
> Determine a region whose area is equal to the given limit. Do not evaluate the limit. lim A" -1 in tan 4n 4n
> Suppose that Sue runs faster than Kathy throughout a 1500-meter race. What is the physical meaning of the area between their velocity curves for the first minute of the race?
> State the rules for approximating the definite integral fbaf (x) dx with the Midpoint Rule, the Trapezoidal Rule, and Simpson’s Rule. Which would you expect to give the best estimate? How do you approximate the error for each rule?
> Explain exactly what is meant by the statement that “differentiation and integration are inverse processes.”
> (a). Given an initial approximation x1 to a root of the equation f (x) = 0, explain geometrically, with a diagram, how the second approximation x2 in Newton’s method is obtained. (b). Write an expression for x2 in terms of x1, f (x1), and f(x1). (c). Wri
> What is a normal distribution? What is the significance of the standard deviation?
> Suppose f (x) is the probability density function for the weight of a female college student, where is measured in pounds. (a). What is the meaning of the integral f1300f (x) dx? (b). Write an expression for the mean of this density function. (c). How ca
> If a rectangle has its base on the -axis and two vertices on the curve y = e-x2, show that the rectangle has the largest possible area when the two vertices are at the points of inflection of the curve.
> (a). What is an antiderivative of a function f? (b). Suppose F1 and F2 are both antiderivatives of f on an interval I. How are F1 and F2 related?
> (a). What is the cardiac output of the heart? (b). Explain how the cardiac output can be measured by the dye dilution method.
> Given a demand function p (x), explain what is meant by the consumer surplus when the amount of a commodity currently available is X and the current selling price is p. Illustrate with a sketch.
> A company modeled the demand curve for its product (in dollars) by the equation Use a graph to estimate the sales level when the selling price is $16. Then find (approximately) the consumer surplus for this sales level. 800,000e/S000 p = x + 20,00
> Use Newton’s method to approximate the given number correct to eight decimal places. 3√20
> The demand function for a certain commodity is p = 20 – 0.05x. Find the consumer surplus when the sales level is 300. Illustrate by drawing the demand curve and identifying the consumer surplus as an area.
> The marginal cost of producing units of a certain product is 74 + 1.1x – 0.002x2 + 0.00004x3 (in dollars per unit). Find the increase in cost if the production level is raised from 1200 units to 1600 units.
> The marginal revenue from the sale of units of a product is 12 – 0.0004x. If the revenue from the sale of the first 1000 units is $12,400, find the revenue from the sale of the first 5000 units.
> After an 8-mg injection of dye, the readings of dye concentration, in mg/L, at two-second intervals are as shown in the table. Use Simpson’s Rule to estimate the cardiac output. c(1) c(t) 12 3.9 2.4 14 2.3 4 5.1 16 1.6 6 7.8 18 0.7
> The dye dilution method is used to measure cardiac output with 6 mg of dye. The dye concentrations, in mg/L, are modeled by c (t) = 20t2-0.6t, 0 < t < 10, where is measured in seconds. Find the cardiac output.
> If n is a positive integer, prove that f10 (ln x) n dx = (-1) n n!
> High blood pressure results from constriction of the arteries. To maintain a normal flow rate (flux), the heart has to pump harder, thus increasing the blood pressure. Use Poiseuille’s Law to show that if R0 and P0 are normal values of
> If revenue flows into a company at a rate of f (t) 9000 √1 + 2t, where is measured in years and f (t) is measured in dollars per year, find the total revenue obtained in the first four years.
> If the amount of capital that a company has at time t is f (t), then the derivative, f'(t), is called the net investment flow. Suppose that the net investment flow is √t million dollars per year (where is measured in years). Find the increase in capital
> A movie theater has been charging $7.50 per person and selling about 400 tickets on a typical weeknight. After surveying their customers, the theater estimates that for every 50 cents that they lower the price, the number of moviegoers will increase by 3
> The marginal cost function C (x) was defined to be the derivative of the cost function. (See Sections 3.8 and 4.6.) If the marginal cost of manufacturing meters of a fabric is C'(x) = 5 – 0.008x + 0.000009x2 (measured in dollars per meter) and the fixed
> Use Newton’s method to find the absolute maximum value of the function f (x) = x cos x, 0 < x < π, correct to six decimal places.
> A spring has natural length 20 cm. Compare the work W1 done in stretching the spring from 20 cm to 30 cm with the work W2 done in stretching it from 30 cm to 40 cm. How are W2 and W1 related?
> If the work required to stretch a spring 1 ft beyond its natural length is 12 ft-lb, how much work is needed to stretch it 9 in. beyond its natural length?
> Suppose that 2 J of work is needed to stretch a spring from its natural length of 30 cm to a length of 42 cm. (a). How much work is needed to stretch the spring from 35 cm to 40 cm? (b). How far beyond its natural length will a force of 30 N keep the spr
> Find the average value of the function on the given interval. h(u) — (3 — 2и) -, [-1,1]
> Does the function f (x) = e10|x-2|-x2 have an absolute maximum? If so, find it. What about an absolute minimum?
> Find the average value of the function on the given interval. h(x) — сos'x sin x, [0, п]
> The table shows values of a force function f (x), where is measured in meters and f (x) in newtons. Use Simpson’s Rule to estimate the work done by the force in moving an object a distance of 18 m. 3 9 12 15 18 f(x) 9.8 9.1 8.5 8.0
> Shown is the graph of a force function (in newtons) that increases to its maximum value and then remains constant. How much work is done by the force in moving an object a distance of 8 m? FA (N) 30 20 10 1 2 3 4 5 6 7 8 (m)
> Find the average value of the function on the given interval. f(x) = sin 4x, [-7, T]
> Use the result of Exercise 65 in Section 5.5 to compute the average volume of inhaled air in the lungs in one respiratory cycle. Exercise 65 in Section 5.5: Breathing is cyclic and a full respiratory cycle from the beginning of inhalation to the end of
> Use Simpson’s Rule with n = 10 to estimate the arc length of the curve. Compare your answer with the value of the integral produced by your calculator. x= y + Vy, 1< y< 2
> The graph of f is shown. Evaluate each integral by interpreting it in terms of areas. y= f(x) -2 6 8 4, 2. (b) f(x) dx (c) f, s(x) dx (d) C f(x) dx
> Show how to approximate the required work by a Riemann sum. Then express the work as an integral and evaluate it. A chain lying on the ground is 10 m long and its mass is 80 kg. How much work is required to raise one end of the chain to a height of 6 m?
> The linear density in a rod 8 m long is 12/√x + 1, kg/m, where x is measured in meters from one end of the rod. Find the average density of the rod.
> If a cup of coffee has temperature 950C in a room where the temperature is 200C, then, according to Newton’s Law of Cooling, the temperature of the coffee after t minutes is T (t) = 20 + 75e-t/50. What is the average temperature of the coffee during the
> Show that x2y2 (4 – x2) (4 – y2) < 16 for all numbers and such that |x| < 2 and |y| < 2.
> Investigate the family of functions f (x) = =cxe – cx2. What happens to the maximum and minimum points and the inflection points as changes? Illustrate your conclusions by graphing several members of the family.
