Question: Use Theorem I to compute the shaded

> Find the area of the region between the curve and the x-axis. f (x) = 1 - x2, from -2 to 2

> Let g(x) be the function pictured in Fig. 26. Determine whether 1 > 0 g(x)dx is positive, negative, or zero. Figure 26: 2 1 -2 -3 y y = g(x) +x + 1 2 3 4 5/6 7 8

> Let f (x) be the function pictured in Fig. 25. Determine whether 1 > 0 f (x)dx is positive, negative, or zero. Figure 25: 3 2 1 -1 Y 12 y = f(x) 3 4 5 7 8 8

> Shade the portion of Fig. 24 whose area is given by the integral 0∫1L [ f (x) - g (x)] + 1∫2 [ g (x) - f (x)] dx. Figure 24: 2 y = f(x) 1 y = g(x) 0 У х а 1 2

> Compute the area of the shaded region in two different ways: (a) by using simple geometric formulas; (b) by applying Theorem I. Y f(x) = -x 2- -2 2 0

> Compute the area of the shaded region in two different ways: (a) by using simple geometric formulas; (b) by applying Theorem I. y 1 0 1 (x)=1 1 2 x

> Compute the area of the shaded region in two different ways: (a) by using simple geometric formulas; (b) by applying Theorem I. 2 Y 0 12 f(x) = 2 23 3 4

> Evaluate a Riemann sum to approximate the area under the graph of f (x) on the given interval, with points selected as specified. f (x) = x√(1 + x2); 1 ≤ x ≤ 3, n = 20, midpoints of subintervals f

> Evaluate a Riemann sum to approximate the area under the graph of f (x) on the given interval, with points selected as specified. f (x) = x√(1 + x2); 1 ≤ x ≤ 3, n = 20, midpoints of subintervals

> Determine the following: ∫ (2/√x + 2√x) dx

> Compute the area under the graph of y = 1 / (1 + x2) from 0 to 5.

> The area under the graph of the function e-x2 plays an important role in probability. Compute this area from -1 to 1.

> We show that, as the number of subintervals increases indefinitely, the Riemann sum approximation of the area under the graph of f (x) = x2 from 0 to 1 approaches the value 1/3 , which is the exact value of the area. Partition the interval [0, 1] into n

> We show that, as the number of subintervals increases indefinitely, the Riemann sum approximation of the area under the graph of f (x) = x2 from 0 to 1 approaches the value 1/3 , which is the exact value of the area. Verify the given formula for n = 1, 2

> A farmer wants to divide the lot in Fig. 18 into two lots of equal area by erecting a fence that extends from the road to the river as shown. Determine the location of the fence. Figure 18: 10 20 30 40 Road 50 60 70 80 06 ft 40 ft 35 ft 30 ft 25 ft

> Estimate the area (in square feet) of the residential lot in Fig. 17. Figure 17: 0 20 60 100 140 160 106 ft 101 ft 100 ft 113 ft

> Use a Riemann sum with n = 5 and midpoints to estimate the area under the graph of f (x) = √(1 - x2) on the interval 0 ≤ x ≤ 1. The graph is a quarter circle, and the area under the graph is .7854

> The graph of the function f (x) = √(1 - x2) on the interval -1 ≤ x ≤ 1 is a semicircle. The area under the graph is 1/2 π(1)2 = π/2 = 1.57080, to five decimal places. Use a R

> Use a Riemann sum with n = 4 and right endpoints to estimate the area under the graph of f (x) = 2x - 4 on the interval 2 ≤ x ≤ 3. Then, repeat with n = 4 and midpoints. Compare the answers with the exact answer, 1

> Use a Riemann sum with n = 4 and left endpoints to estimate the area under the graph of f (x) = 4 - x on the interval 1 ≤ x ≤ 4. Then repeat with n = 4 and midpoints. Compare the answers with the exact answer, 4.5,

> Determine the following: ∫x√x dx

> Use a Riemann sum to approximate the area under the graph of f (x) in Fig. 14 on the given interval, with selected points as specified. Draw the approximating rectangles. 1 ≤ x ≤ 7, n = 3, midpoints of subintervals

> Use a Riemann sum to approximate the area under the graph of f (x) in Fig. 14 on the given interval, with selected points as specified. Draw the approximating rectangles. 4 ≤ x ≤ 9, n = 5, right endpoints Riemann

> Use a Riemann sum to approximate the area under the graph of f (x) in Fig. 14 on the given interval, with selected points as specified. Draw the approximating rectangles. 3 ≤ x ≤ 7, n = 4, left endpoints Riemann s

> Use a Riemann sum to approximate the area under the graph of f (x) in Fig. 14 on the given interval, with selected points as specified. Draw the approximating rectangles. 0 ≤ x ≤ 8, n = 4, midpoints of subintervals

> Use a Riemann sum to approximate the area under the graph of f (x) on the given interval, with selected points as specified. f (x) = ln x; 2 ≤ x ≤ 4, n = 5, left endpoints Riemann sum: Ricmann sum. f(x1) Δx +

> Use a Riemann sum to approximate the area under the graph of f (x) on the given interval, with selected points as specified. f (x) = e-x; 2 ≤ x ≤ 3, n = 5, right endpoints Riemann sum: Ricmann sum. f(x1) Δx +

> Use a Riemann sum to approximate the area under the graph of f (x) on the given interval, with selected points as specified. f (x) = x3; 0 ≤ x ≤ 1, n = 5, right endpoints Riemann sum: Ricmann sum. f(x1) Δx +

> Use a Riemann sum to approximate the area under the graph of f (x) on the given interval, with selected points as specified. f (x) = x3; 1 ≤ x ≤ 3, n = 5, left endpoints Riemann sum: Ricmann sum. f(x1) Δx + f

> Use a Riemann sum to approximate the area under the graph of f (x) on the given interval, with selected points as specified. f (x) = x2; -2 ≤ x ≤ 2, n = 4, midpoints of subintervals Riemann sum: Ricmann sum.

> Use a Riemann sum to approximate the area under the graph of f (x) on the given interval, with selected points as specified. f (x) = x2; 1 ≤ x ≤ 3, n = 4, midpoints of subintervals Riemann sum: Ricmann sum. f

> Determine the following: ∫1/7x dx

> Determine Δx and the midpoints of the subintervals formed by partitioning the given interval into n subintervals. 3 ≤ x ≤ 5; n = 5

> Determine Δx and the midpoints of the subintervals formed by partitioning the given interval into n subintervals. 1 ≤ x ≤ 4; n = 5

> Determine Δx and the midpoints of the subintervals formed by partitioning the given interval into n subintervals. 0 ≤ x ≤ 3; n = 6

> Determine Δx and the midpoints of the subintervals formed by partitioning the given interval into n subintervals. 0 ≤ x ≤ 2; n = 4

> Find the real number b 7 0 so that the area under the graph of y = x2 from 0 to b is equal to the area under the graph of y = x3 from 0 to b.

> Find the real number b > 0 so that the area under the graph of y = x3 from 0 to b is equal to 4.

> Find the area under each of the given curves. y = e3x; x = - 1/3 to x = 0

> Find the area under each of the given curves. y = (x - 3)4; x = 1 to x = 4

> Find the area under each of the given curves. y = √x; x = 0 to x = 4

> Find the area under each of the given curves. y = 3x2 + x + 2ex/2; x = 0 to x = 1

> Determine the following: ∫ (2/x + x/2) dx

> Find the area under each of the given curves. y = 3x2; x = -1 to x = 1

> Find the area under each of the given curves. y = 4x; x = 2 to x = 3

> Draw the region whose area is given by the definite integral. 0∫4√x dx

> Draw the region whose area is given by the definite integral. 0∫4 (8 - 2x) dx

> Draw the region whose area is given by the definite integral. 2∫4 x2 dx

> Use Theorem I to compute the shaded area in Exercise 11. Shaded area in Exercise 11: Theorem 1: y y = x + 1 1 3 y=x Theorem I: Area under a Graph If f(x) is a continuous nonnegative function on the interval a ≤ x ≤ b, then the area under the grap

> Use Theorem I to compute the shaded area in Exercise 7. Shaded Area in Exercise 7: Theorem 1: Y 0 f(x)=1/ 1 2 Theorem I: Area under a Graph If f(x) is a continuous nonnegative function on the interval a ≤ x ≤ b, then the area under the graph of f

> Set-up the definite integral that gives the area of the shaded region. Do not evaluate the integral. y 2. 1 0 x+1 I 1 1 3-x 2

> Set-up the definite integral that gives the area of the shaded region. Do not evaluate the integral. y y = x + ² 1 3 y=x

> Determine the following: ∫x * x2 dx

> Set-up the definite integral that gives the area of the shaded region. Do not evaluate the integral. Y -1 이 y=e 2

> Set-up the definite integral that gives the area of the shaded region. Do not evaluate the integral. Y f(x) = ln x 1 2

> Set-up the definite integral that gives the area of the shaded region. Do not evaluate the integral. fi 0 y = -x(x-3) 3 x

> Set-up the definite integral that gives the area of the shaded region. Do not evaluate the integral. Y 0 f(x)=1/ 1 2 x

> Compute the area of the shaded region in two different ways: (a) by using simple geometric formulas; (b) by applying Theorem I. 2. g 1 2 3 = 6 - 2x 3 - Ꮖ

> Compute the area of the shaded region in two different ways: (a) by using simple geometric formulas; (b) by applying Theorem I. 1 y 0 y = 1- x 1 y = x - 1 2

> Compute the area of the shaded region in two different ways: (a) by using simple geometric formulas; (b) by applying Theorem I. -2 22 Y 0 f(x) = x + 2 2

> Evaluate the given integral. 1∫4 (3√t + 4t) dt

> Evaluate the given integral. -1∫2(x2/3 – 2/9x) dx

> Evaluate the given integral. 0∫1 (2x – 3/4) dx

> Determine the following: ∫x/c dx (c a constant ≠ 0)

> A conical-shaped tank is being drained. The height of the water level in the tank is decreasing at the rate h(t) = - t/2 inches per minute. Find the decrease in the depth of the water in the tank during the time interval 2 ≤ t ≤ 4.

> A saline solution is being flushed with fresh water in such a way that salt is eliminated at the rate r(t) = -(t + ½) grams per minute. Find the amount of salt that is eliminated during the first 2 minutes.

> A sample of radioactive material with decay constant .1 is decaying at a rate R(t) = -e-0.1t grams per year. How many grams of this material decayed after the first 10 years?

> Using the data from the previous exercise, find P(t). Exercise 40: You took a $200,000 home mortgage at an annual interest rate of 3%. Suppose that the loan is amortized over a period of 30 years, and let P(t) denote the amount of money (in thousands of

> You took a $200,000 home mortgage at an annual interest rate of 3%. Suppose that the loan is amortized over a period of 30 years, and let P(t) denote the amount of money (in thousands of dollars) that you owe on the loan after t years. A reasonable estim

> The rate of change of a population with emigration is given by P(t) = 7/300 et/25 – 1/80 et/16, where P(t) is the population in millions, t years after the year 2000. (a) Estimate the change in population as t varies from 2000 to 2010. (b) Estimate the

> A property with an appraised value of $200,000 in 2015 is depreciating at the rate R(t) = -8e-0.04t, where t is in years since 2015 and R(t) is in thousands of dollars per year. Estimate the loss in value of the property between 2015 and 2021 (as t varie

> An investment grew at an exponential rate R(t) = 700e0.07t + 1000, where t is in years and R(t) is in dollars per year. Approximate the net increase in value of the investment after the first 10 years (as t varies from 0 to 10).

> A company’s marginal cost function is given by C(x) = 32 + x/20, where x denotes the number of items produced in 1 day and C(x) is in thousands of dollars. Determine the increase in cost if the company goes from a production level of 15 to 20 items per

> A company’s marginal cost function is .1x2 - x + 12 dollars, where x denotes the number of units produced in 1 day. (a) Determine the increase in cost if the production level is raised from x = 1 to x = 3 units. (b) If C(1) = 15, determine C(3) using you

> Determine the following: ∫k2 dx (k a constant)

> Find all antiderivatives of each following function: f (x) = x

> If f (x) and g (x) are differentiable functions, find g (x) if you know that d/dx f ( g (x)) = 3x2 * f (x3 + 1).

> A manufacturer of microcomputers estimates that t months from now it will sell x thousand units of its main line of microcomputers per month, where x = .05t2 + 2t + 5. Because of economies of scale, the profit P from manufacturing and selling x thousand

> Ecologists estimate that, when the population of a certain city is x thousand persons, the average level L of carbon monoxide in the air above the city will be L ppm (parts per million), where L = 10 + .4x + .0001x2. The population of the city is estimat

> The cost of manufacturing x cases of cereal is C dollars, where C = 3x + 4√x + 2. Weekly production at t weeks from the present is estimated to be x = 6200 + 100t cases. (a) Find the marginal cost, dC/dx. (b) Find the time rate of change of cost, dC/dt.

> When a company produces and sells x thousand units per week, its total weekly profit is P thousand dollars, where P = 200x / (100 + x2). The production level at t weeks from the present is x = 4 + 2t. (a) Find the marginal profit, dP/dx. (b) Find the t

> Suppose that Q, x, and y are variables, where Q is a function of x and x is a function of y. (Read this carefully.) (a) Write the derivative symbols for the following quantities: (i) the rate of change of x with respect to y; (ii) the rate of change of

> Differentiate the function. y = 1 / (√x – 2)

> Suppose that P, y, and t are variables, where P is a function of y and y is a function of t. (a) Write the derivative symbols for the following quantities: (i) the rate of change of y with respect to t; (ii) the rate of change of P with respect to y;

> Many relations in biology are expressed by power functions, known as allometric equations, of the form y = kxa, where k and a are constants. For example, the weight of a male hognose snake is approximately 446x3 grams, where x is its length in meters. If

> The length, x, of the edge of a cube is increasing. (a) Write the chain rule for dV/dt, the time rate of change of the volume of the cube. (b) For what value of x is dV/dt equal to 12 times the rate of increase of x?

> Radium 226 is used in cancer radiotherapy. Let P(t) be the number of grams of radium 226 in a sample remaining after t years, and let P(t) satisfy the differential equation P(t) = -.00043P(t), P(0) = 12. (a) Find the formula for P(t). (b) What was the i

> The function f (x) = √(x2 - 6x + 10) has one relative minimum point for x ≥ 0. Find it.

> Find the x-coordinates of all points on the curve y = (-x2 + 4x - 3)3 with a horizontal tangent line.

> Find the equation of the line tangent to the graph of y = x/(√2 - x2) at the point (1, 1).

> Find the equation of the line tangent to the graph of y = 2x(x - 4)6 at the point (5, 10).

> Compute dy/dt |t=t0. y = √(x + 1), x = √(t + 1), t0 = 0

> Compute dy/dt |t=t0. y = (x + 1)/(x – 1), x = t2/4, t0 = 3

> Compute dy/dt |t=t0. y = (x2 - 2x + 4)2, x = 1/t + 1, t0 = 1

> Differentiate the function. y = [(3x2 + 2x + 2)(x - 2)]2

> Compute dy/dt |t=t0. y = x2 - 3x, x = t2 + 3, t0 = 0

> Compute dy/dx using the chain rule in formula (1). State your answer in terms of x only. y = (u2 + 2u)/(u + 1), u = x(x + 1) Chain Rule: The Chain Rule To differentiate f(g(x)), differentiate first the outside function f(x) and substitute g(x) for

> A sample of 8 grams of radioactive material is placed in a vault. Let P(t) be the amount remaining after t years, and let P(t) satisfy the differential equation P(t) = -.021P(t). (a) Find the formula for P(t) (b) What is P(0)? (c) What is the decay cons

> Compute dy/dx using the chain rule in formula (1). State your answer in terms of x only. y = u/2 + 2/u , u = x - x2 Chain Rule: The Chain Rule To differentiate f(g(x)), differentiate first the outside function f(x) and substitute g(x) for x in the

> Compute dy/dx using the chain rule in formula (1). State your answer in terms of x only. y = √(u + 1), u = 2x2 Chain Rule: The Chain Rule To differentiate f(g(x)), differentiate first the outside function f(x) and substitute g(x)