Find the volume of the solid of revolution generated by revolving about the x-axis the region under the curve y = 1 - x2 from x = 0 to x = 1.
> A rock is dropped from the top of a 400-foot cliff. Its velocity at time t seconds is y(t) = -32t feet per second. (a) Find s(t), the height of the rock above the ground at time t. (b) How long will the rock take to reach the ground? (c) What will be its
> A ball is thrown upward from a height of 256 feet above the ground, with an initial velocity of 96 feet per second. From physics it is known that the velocity at time t is y(t) = 96 - 32t feet per second. (a) Find s(t), the function giving the height abo
> The function g(x) in Fig. 9 resulted from shifting the graph of f (x) up 2 units. What is the derivative of h(x) = g(x) - f (x)? Figure 9: Y + 2 y = g(x) y = f(x) X
> The function g(x) in Fig. 8 resulted from shifting the graph of f (x) up 3 units. If f ‘(5) = 1/4 , what is g(5)? Figure 8: 3 ↓ 5 y = g(x) y = f(x) x
> Figure 7 contains an antiderivative of the function f (x). Draw the graph of another antiderivative of f (x). Figure 7: TH Y [T I
> Figure 6 contains the graph of a function F (x). On the same coordinate system, draw the graph of the function G(x) having the properties G (0) = 0 and G ’(x) = F ‘(x) for each x. Figure 6: 2 -2 y H 4 y = F(x) 8
> Which of the following is ∫x√(x + 1) dx? (a) 2/5 (x + 1)5/2 – 2/3 (x + 1)3/2 + C (b) ½ x2 * 2/3 (x + 1)3/2 + C
> Find all antiderivatives of each following function: f (x) = 3
> Which of the following is ∫ln x dx? (a) 1/x + C (b) x * ln x - x + C (c) ½ * (ln x)2 + C
> Figure 5 shows the graphs of several functions f (x) for which f ‘(x) = 1/3. Find the expression for the function f (x) whose graph passes through (6, 3). Figure 5: ∞ 6 4 Y + 2 (6,3) 6 8 8 10 -x 12
> Figure 4 shows the graphs of several functions f (x) for which f (x) = 2/x. Find the expression for the function f ‘(x) whose graph passes through (1, 2). Figure 4: (1, 2) 6 4 2 y -2. -4- 4 6 + 8 | | | 10 12 X
> Find all functions f (x) that satisfy the given conditions. f ‘(x) = x2 + √x, f (1) = 3
> Find all functions f (x) that satisfy the given conditions. f ‘(x) = √x + 1, f (4) = 0
> Find all functions f (x) that satisfy the given conditions. f ‘ (x) = 8x1/3, f (1) = 4
> Find all functions f (x) that satisfy the given conditions. f ‘(x) = x, f (0) = 3
> Find all functions f (x) that satisfy the given conditions. f ‘(x) = 2x - e-x, f (0) = 1
> Find all functions f (x) that satisfy the given conditions. f ‘(x) = .5e-0.2x, f (0) = 0
> Find all functions f (t) that satisfy the given condition. f ‘(t) = t2 - 5t - 7
> Find all antiderivatives of each following function: f (x) = e-3x
> Find all functions f (t) that satisfy the given condition. f ‘(t) = 0
> Find all functions f (t) that satisfy the given condition. f ‘(t) = 4/(6 + t)
> Find all functions f (t) that satisfy the given condition. f ‘(t) = t3/2
> Find the value of k that makes the antidifferentiation formula true. ∫5/(2 - 3x) dx = k ln |2 - 3x| + C
> Find the value of k that makes the antidifferentiation formula true. ∫3/(2 + x) dx = k ln |2 + x| + C
> Calculate the following integrals. ∫ (x3 + 3x2 - 1)dx
> Calculate the following integrals. ∫ 2/(x + 4) dx
> Calculate the following integrals. ∫ 2x + 1 dx
> Calculate the following integrals. ∫ (x2 - 3x + 2) dx
> Calculate the following integrals. ∫32 dx
> Generalize the result of Exercise 73 as follows: Let n be a positive integer. Show that 0∫1 (n√x – xn) dx = (n-1)/(n+1).
> Find the value of k that makes the antidifferentiation formula true. ∫ (2x - 1)3 dx = k(2x - 1)4 + C
> Show that 0∫1 (√x - x2) dx = 1/3.
> Generalize the result of Exercise 71 as follows: Let n be a positive integer. Show that for any positive number b we have 0∫bn n√x dx + 0∫b xn dx = bn+1
> Show that for any positive number b we have 0∫b2 √x dx + 0∫b x2 dx = b3
> For what value of a is the shaded area in Fig. 4 equal to 1? Figure 4: y = r ย 0 D a, a
> Find a function f (x) whose graph goes through the point (1, 1) and whose slope at any point (x, f (x)) is 3x2 - 2x + 1.
> If money is deposited steadily in a savings account at the rate of $4500 per year, determine the balance at the end of 1 year if the account pays 9% interest compounded continuously.
> The annual world rate of water use t years after 1960, for t ≤ 35, was approximately 860 e0.04t cubic kilometers per year. How much water was used between 1960 and 1995?
> Suppose that water is flowing into a tank at a rate of r (t) gallons per hour, where the rate depends on the time t according to the formula r (t) = 20 - 4t, 0 ≤ t ≤ 5. (a) Consider a brief period of time, say, from t1 to t2. The length of this time peri
> True or false: If 3 ≤ f (x) ≤ 4 whenever 0 ≤ x ≤ 5, then 3 ≤ 1/5 0∫5 f (x) dx ≤ 4.
> In Fig. 3, the rectangle has the same area as the region under the graph of f (x). What is the average value of f (x) on the interval 2 ≤ x ≤ 6? Figure 3: fi 0 2 6 x y 0 y = f(x) 2 6
> Find the value of k that makes the antidifferentiation formula true. ∫ (3x + 2)4 dx = k(3x + 2)5 + C
> What number does the sum [13 + (1 + 1/n)3 + (1 + 2/n)3+ (1 + 3/n)3+ … + (1 + n-1/n)3] * 1/n approach as n gets very large?
> What number does [e0 + e1/n + e2/n + e3/n + … + e(n-1)/n] * 1/n approach as n gets very large?
> Suppose that the interval 0 ≤ t ≤ 3 is divided into 1000 subintervals of width Δt. Let t1, t2, … , t1000 denote the right endpoints of these subintervals. If we need to estimate the
> For each number x satisfying -1 … x … 1, define h(x) by h(x) = -1∫x √ (1 - t2) dt. (a) Give a geometric interpretation of the values h(0) and h(1). (b) Find the derivative h ‘(x).
> Let x be any positive number, and define g(x) to be the number determined by the definite integral g(x) = 0∫x 1/(1 + t2) dt. (a) Give a geometric interpretation of the number g(3). (b) Find the derivative g ‘ (x).
> A retail store sells a certain product at the rate of g(t) units per week at time t, where g(t) = rt. At time t = 0, the store has Q units of the product in inventory. (a) Find a formula f (t) for the amount of product in inventory at time t. (b) Determi
> A store has an inventory of Q units of a certain product at time t = 0. The store sells the product at the steady rate of Q/A units per week and exhausts the inventory in A weeks. (a) Find a formula f (t) for the amount of product in inventory at time t.
> In Fig. 2, three regions are labeled with their areas. Determine a∫c f (x) dx and determine a∫d f (x) dx. Figure 2: y a .68 b .42 1.7 d
> Suppose that the interval 0 ≤ x ≤ 1 is divided into 100 subintervals with a width of Δx = .01. Show that the sum [3e-0.01] Δx + [3e-0.02] Δx + [3e-0.03] Δx + … + [3e-1] Δx is close to 3(1 - e-1).
> Find the value of k that makes the antidifferentiation formula true. ∫7/(8 - x)4 dx = k/(8 - x)3 + C
> Find the average value of f (x) = 1/x3 from x = 1/3 to x = 1/2.
> Three thousand dollars is deposited in the bank at 4% interest compounded continuously. What will be the average value of the money in the account during the next 10 years?
> Find the consumers’ surplus for the demand curve p = √(25 - .04x) at the sales level x = 400.
> Use a Riemann sum with n = 5 and midpoints to estimate the area under the graph of f (x) = e2x on the interval 0 ≤ x ≤ 1. Then, use a definite integral to find the exact value of the area to five decimal places.
> Use a Riemann sum with n = 2 and midpoints to estimate the area under the graph of f (x) = 1/(x + 2) on the interval 0 ≤ x ≤ 2. Then, use a definite integral to find the exact value of the area to five decimal places.
> Redo Exercise 47 using right endpoints. Exercise 47: Use a Riemann sum with n = 4 and left endpoints to estimate the area under the graph in Fig. 1 for 0 ≤ x ≤ 2. Figure 1: 20 10 ม 0 (.5, 14) .5 (1, 10) 1 (1.
> Use a Riemann sum with n = 4 and left endpoints to estimate the area under the graph in Fig. 1 for 0 ≤ x ≤ 2. Figure 1: 20 10 ม 0 (.5, 14) .5 (1, 10) 1 (1.5, 6) 1.5 (2,4) 2
> A rock thrown straight up into the air has a velocity of υ(t) = -9.8t + 20 meters per second after t seconds. (a) Determine the distance the rock travels during the first 2 seconds. (b) Represent the answer to part (a) as an area.
> A drug is injected into a patient at the rate of f (t) cubic centimeters per minute at time t. What does the area under the graph of y = f (t) from t = 0 to t = 4 represent?
> If the marginal revenue function for a company is 400 - 3x2, find the additional revenue received from doubling production if 10 units are currently being produced.
> Find the value of k that makes the antidifferentiation formula true. ∫ (4 - x)-1 dx = k ln |4 – x| + C
> An airplane tire plant finds that its marginal cost of producing tires is .04x + $150 at a production level of x tires per day. If fixed costs are $500 per day, find the cost of producing x tires per day.
> Let k be a constant, and let y = f (t) be a function such that y ’ = kty. Show that y = Cekt2/2, for some constant C.
> Describe all solutions of the following differential equations, where y represents a function of t. (a) y ’ = 4t (b) y ’ = 4y (c) y ’ = e4t
> Find the function f (x) for which f ‘(x) = e-5x, f (0) = 1.
> Find the function f (x) for which f ‘ (x) = (x - 5)2, f (8) = 2.
> Find the area of the region between the curves y = 2x2 + x and y = x2 + 2 from x = 0 to x = 2.
> Find the area of the region bounded by the curves y = x3 - 3x + 1 and y = x + 1.
> Find the area of the shaded region. -2 ม y = 2 – 2 – in 3
> Find the area of the shaded region. Y 0 y=e² - ex 1
> Find the area of the shaded region. Y 0 1/2 y=x²-x-1 y = 1-1/x 1 2
> Find the value of k that makes the antidifferentiation formula true. ∫√(x + 1) dx = k(x + 1)3/2 + C
> Find all antiderivatives of each following function: f (x) = e3x
> Find the area of the shaded region. 4 0 y=4-2² 1 y=1-x² 2
> Find the area of the shaded region. 1.21 3 = fi X} = f 0 "
> Find the area of the shaded region. y=e-z Y 0 1 y=e² In 2 1
> Find the area of the shaded region. -2 =R Y 0 y = x³ + 2x 2
> Find the area of the shaded region. x x=k xt = f 0 fi
> Find the area under the curve y = 1 + √x from x = 1 to x = 9.
> Find the area under the curve y = (3x - 2)-3 from x = 1 to x = 2.
> Calculate the following integrals. 0∫1 (3 + e2x)/ex dx
> Calculate the following integrals. 0∫ln 3 (ex + e-x)/ e2x dx
> Calculate the following integrals. ln 2∫ln 3 (ex + e-x) dx
> Find the value of k that makes the antidifferentiation formula true. ∫ (5x - 7)-2 dx = k(5x - 7)-1 + C
> Calculate the following integrals. 0∫ln 2 (ex - e-x) dx
> Calculate the following integrals. -2∫2 3/2e3x dx
> Calculate the following integrals. 0∫5 (5 + 3x)-1 dx
> Calculate the following integrals. 3∫6 e2-(x/3) dx
> Calculate the following integrals. 1∫4 1/x2 dx
> Calculate the following integrals. 2/3 0∫8 √ (x + 1) dx
> Calculate the following integrals. 1∫24/x5 dx
> Calculate the following integrals. 0∫1 [2/(x + 1) – 1/(x + 4)] dx
> Calculate the following integrals. -1∫2 √ (2x + 4) dx
> Calculate the following integrals. 0∫1/8 5√x dx
> Find the value of k that makes the antidifferentiation formula true. ∫4 e3x+1 dx = k e3x+1 + C
> Calculate the following integrals. -1∫1 (x + 1)2 dx
> Calculate the following integrals. ∫ (5/x – x/5) dx
> Calculate the following integrals. ∫ √(4 – x) dx
