If f is a continuous function, what is the limit as f→0 of the average value of f on the interval [x, x + h]?
> If n is a positive integer, prove that f10 (ln x) n dx = (-1) n n!
> If f' is continuous on [a, b], show that 2 fba f (x) f' (x) dx = [f (b)]2 – [f (a)]2
> Find a function f and a value of the constant a such that 2 f(t) dt = 2 sin x – 1
> Let be the region bounded by the curves y = tan (x2), x = 1 and y = 0. Use the Midpoint Rule with n = 4 to estimate the following quantities. (a). The area of (b). The volume obtained by rotating about the -axis
> If f is a continuous function such that for all x, find an explicit formula for f (x). , r) dt = xe² + [ e "f)di
> Find the local and absolute extreme values of the function on the given interval. f(x) = x' – 6x² + 9x + 1, [2, 4]
> Suppose that the temperature in a long, thin rod placed along the x-axis is initially C/(2a) if |x| a. It can be shown that if the heat diffusivity of the rod is k, then the temperature of the rod at the point x at time t is To find the temperature d
> A population of honeybees increased at a rate of r (t) bees per week, where the graph of is as shown. Use Simpson’s Rule with six subintervals to estimate the increase in the bee population during the first 24 weeks. 12000 8000 400
> Let r (t) be the rate at which the world’s oil is consumed, where is measured in years starting at t = 0 on January 1, 2000, and r (t) is measured in barrels per year. What does f80r (t) dt represent?
> The speedometer reading (v) on a car was observed at 1-minute intervals and recorded in the chart. Use Simpson’s Rule to estimate the distance traveled by the car. t (min) v (mi/h) t (min) v (mi/h) 40 6 56 1 42 7 57 2 45 8 57 3 49
> Evaluate the integral. f01 x4/5 dx
> (a). Evaluate the integral f∞0 xne-x dx for n = 0, 1, 2, and 3. (b). Guess the value of f∞0 xne-x dx when n is an arbitrary positive integer. (c). Prove your guess using mathematical induction.
> A particle moves along a line with velocity function v (t) = t2 - t, where is measured in meters per second. Find (a) the displacement and (b) the distance traveled by the particle during the time interval [0, 5].
> For what values of a is f∞0eax cos x dx convergent? Use the Table of Integrals to evaluate the integral for those values of a.
> Use the Comparison Theorem to determine whether the Integral is convergent or divergent. dx x' + 2
> Evaluate the integral or show that it is divergent. f62y/√y – 2, dy
> Find the volume of the solid obtained by rotating about the x-axis the region bounded by the curves y = e-2x, y = 1 + x and x = 1.
> Evaluate the integral or show that it is divergent. fe1dx/x √lnx
> Evaluate the integral or show that it is divergent. f101/2 – 3x, dx
> Evaluate the integral or show that it is divergent. f0-∞e-2x dx
> Evaluate the integral or show that it is divergent. f∞0ln x/x4, dx
> Use a graph to give a rough estimate of the area of the region that lies beneath the given curve. Then find the exact area. y = sin x, 0 <I<T
> Evaluate the integral or show that it is divergent. f∞11/ (2x + 1)3, dx
> Use Newton’s method to find the coordinates, correct to six decimal places, of the point on the parabola y = (x – 1)2 that is closest to the origin.
> Use the properties of integrals to verify that 0 < f10x4 cos x dx < 0.2.
> Use Property 8 of integrals to estimate the value of f31√x2 + 3, dx.
> Find the most general antiderivative of the function. g(1) = (1 + t)//E
> Find the most general antiderivative of the function. f(x) = e* – (2//x)
> Use Simpson’s Rule with n = 6 to estimate the area under the curve y = ex/x from x = 1 to x = 4.
> The curve traced out by a point at a distance 1 m from the center of a circle of radius 2 m as the circle rolls along the x-axis is called a trochoid and has parametric equations One arch of the trochoid is given by the parameter interval 0 x= 20 -
> Use Newton’s method to find the absolute maximum value of the function f (t) = cos t + t – t2 correct to eight decimal places.
> Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson’s Rule with n = 10 to approximate the given integral. Round your answers to six decimal places. Can you say whether your answers are underestimates or overestimates?
> What is wrong with the equation? f0π sec2x dx = tan x]0π = 0
> Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson’s Rule with n = 10 to approximate the given integral. Round your answers to six decimal places. Can you say whether your answers are underestimates or overestimates?
> Use the Table of Integrals on the Reference Pages to evaluate the integral. cot x dx V1 + 2 sin x
> A stone was dropped off a cliff and hit the ground with a speed of 120 ft/s. What is the height of the cliff?
> Use the Table of Integrals on the Reference Pages to evaluate the integral. x² + x + 1 dx
> Use the Table of Integrals on the Reference Pages to evaluate the integral. | csc*t dt
> Use the Table of Integrals on the Reference Pages to evaluate the integral. VT- e2* dx
> Find the derivative of the function. y = f3x+12x sin (t4) dt
> Find the derivative of the function. y = fx√x, et/t, dt
> Find the derivative of the function. g (x) = fsinx11 – t2/1 + t4 dt
> Find the area of the region bounded by the given curves. x + y = 0, x= y² + 3y
> What is wrong with the equation? f-13 1/x2 dx = x-1/-1]-13 = -4/3
> The length of time spent waiting in line at a certain bank is modeled by an exponential density function with mean 8 minutes. (a). What is the probability that a customer is served in the first 3 minutes? (b). What is the probability that a customer has
> Lengths of human pregnancies are normally distributed with mean 268 days and standard deviation 15 days. What percentage of pregnancies last between 250 days and 280 days?
> (a). Explain why the function is a probability density function. (b). Find P (X (c). Calculate the mean. Is the value what you would expect? sin f(x) = { 20 10 if 0 <x< 10 if x<0 or x> 10
> (a). Use Newton’s method with x1 = 1 to find the root of the equation x3 – x = 1 correct to six decimal places. (b). Solve the equation in part (a) using x1 = 0.6 as the initial approximation. (c). Solve the equation in part (a) using x1 = 0.57. (You def
> After a 6-mg injection of dye into a heart, the readings of dye concentration at two-second intervals are as shown in the table. Use Simpson’s Rule to estimate the cardiac output. c(t) c(t) t 14 4.7 1.9 16 3.3 4 3.3 18 2.1 5.1 20 1
> Find the average value of the function f (x) = x2 √1 + x3 on the interval [0, 2].
> The demand function for a commodity is given by p = 200 – 0.1x – 0.01x2 Find the consumer surplus when the sales level is 100.
> Find the centroid of the region shown. yA (3, 2)
> A gate in an irrigation canal is constructed in the form of a trapezoid 3 ft wide at the bottom, 5 ft wide at the top, and 2 ft high. It is placed vertically in the canal so that the water just covers the gate. Find the hydrostatic force on one side of t
> Evaluate the integral. f03π/2 |sin x| dx
> A trough is filled with water and its vertical ends have the shape of the parabolic region in the figure. Find the hydrostatic force on one end of the trough. 8 ft 4 ft
> Find the area of the region bounded by the given curves. y = 1 - 2x², y = |x|
> Evaluate the integral. f dt/t2 + 6t+ 8
> Evaluate the integral. f sin x cos (cos x) dx
> For what values of m do the line y = mx and the curve y = x/ (x2 + 1) enclose a region? Find the area of the region.
> A force of 30 N is required to maintain a spring stretched from its natural length of 12 cm to a length of 15 cm. How much work is done in stretching the spring from 12 cm to 20 cm?
> Find the length of the curve y = fx1√√t, - 1, dt, 1 < x < 16.
> Find the length of the curve y = 1/6 (x2 + 4)3/2, 0 < x < 3.
> Use Simpson’s Rule with n = 10 to estimate the length of the arc of the curve y = 1/x? from (1, 1) to (2, 4).
> Find the length of the curve with parametric equations x= 3t2, y = 2t3, 0 < t < 2.
> Evaluate the integral. f02 (x4 – 3/4x2 + 2/3x -1) dx
> Evaluate the integral. f21 x3 lnx dx
> The height of a monument is 20 m. A horizontal cross section at a distance meters from the top is an equilateral triangle with side 1/4x meters. Find the volume of the monument.
> The base of a solid is the region bounded by the parabolas y = x2 and y = 2 – x2. Find the volume of the solid if the cross-sections perpendicular to the -axis are squares with one side lying along the base.
> Find the area of the region bounded by the given curves. y = 1/x, y = x², y= 0, x= e
> The base of a solid is a circular disk with radius 3. Find the volume of the solid if parallel cross-sections perpendicular to the base are isosceles right triangles with hypotenuse lying along the base.
> Determine a region whose area is equal to the given limit. Do not evaluate the limit. 10 lim 2 A" i-1 n 2i 5 +
> The “Garbage Project” at the University of Arizona reports that the amount of paper discarded by households per week is normally distributed with mean 9.4 lb and standard deviation 4.2 lb. What percentage of households throw out at least 10 lb of paper a
> Suppose you are asked to estimate the volume of a football. You measure and find that a football is 28 cm long. You use a piece of string and measure the circumference at its widest point to be 53 cm. The circumference 7 cm from each end is 45 cm. Use Si
> Describe the solid whose volume is given by the integral. /2 (a) [* 27 cos?x dx 2т сos?x dx (b) 7[(2 – x²)? – (2 – JI):] dx or
> Let R be the region bounded by the curves y = 1 – x2 and y = x6 – x + 1. Estimate the following quantities. (a). The x-coordinates of the points of intersection of the curves (b). The area of R (c). The volume generated when R is rotated about the x-axis
> Evaluate the integral. f-12 (x – 2|x|) dx
> Evaluate the integral. f21 (8x3 + 3x2) dx
> Evaluate: (a). f10d/dx (earctan x) dx (b). d/dx f10 earctan x dx (c). d/dx fx0 earctan t dx
> (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
> Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. y = cos?x, |x|< n/2, y = about x = T/ T/2
> (a). Write f20e3x dx as a limit of Riemann sums, taking the sample points to be right endpoints. Use a computer algebra system to evaluate the sum and to compute the limit. (b). Use the Evaluation Theorem to check your answer to part (a).
> If f60f (x) dx = 10 and f40f (x) dx = 4, find f64f (x) dx.
> Express as a definite integral on the interval [0, π] and then evaluate the integral. lim 2 sin x; Ax
> Evaluate the integral. f 10/ (x – 1) (x2 + 9), dx
> Find the derivative of the function. F (x) = fx0t2/1 + t2, dt
> Graph the function f (x) = cos2x sin3x and use the graph to guess the value of the integral f2π0f (x) dx. Then evaluate the integral to confirm your guess.
> The Fresnel function was defined in Example 4 and graphed in Figures 6 and 7. Figures 6: Figures 7: (a). At what values of does this function have local maxi mum values? (b). On what intervals is the function concave upward? (c). Use a graph to solv
> Let g (x) = f 9x2), where f is twice differentiable for all x, f'(x) > 0 for all x ≠ 0, and f is concave downward on (-∞, 0) and concave upward on (0, ∞). (a). At what numbers does g have an extreme value? (b). Discuss the concavity of g.
> Use a graph to give a rough estimate of the area of the region that lies under the curve y = x√x, 0 < x < 4. Then find the exact area.
> Evaluate the indefinite integral. Illustrate and check that your answer is reasonable by graphing both the function and its antiderivative (take C = 0). dx Vx? + 1
> Evaluate the indefinite integral. Illustrate and check that your answer is reasonable by graphing both the function and its antiderivative (take C = 0). cos x V1 + sin x Sin X
> Evaluate the integral. f10 ex/1 + e2x, dx
> Evaluate the integral. f sec θ tan θ/1 + sec θ, dθ
> Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. y = Vĩ, y = x; about y = 2
> Evaluate the integral. f tan-1 x dx
> Evaluate the integral. f e3√x dx
> (a). Find the Riemann sum for f (x) = sin x, 0 < x < 3π/2, with six terms, taking the sample points to be right endpoints. (Give your answer correct to six decimal places.) Explain what the Riemann sum represents with the aid of a sketch. (b). Repeat par
> 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.
