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 initial amount? (c) What is the decay constant? (d) Approximately how much of the radium will remain after 943 years? (e) How fast is the sample disintegrating when just 1 gram remains? Use the differential equation. (f) What is the weight of the sample when it is disintegrating at the rate of .004 gram per year? (g) The radioactive material has a half-life of about 1612 years. How much will remain after 1612 years? 3224 years? 4836 years?
> 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 8. Shaded area in Exercise 8: Theorem 1: 0 थ्र y = - 0(0 – 3) 3 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 gra
> 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?
> 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)
> Compute dy/dx using the chain rule in formula (1). State your answer in terms of x only. y = u3/2, u = 4x + 1 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 result
> Compute d/dx f (g (x)), where f (x) and g (x) are: f (x) = x(x - 2)4, g (x) = x3
> Solve the following equations for x 22x – 4*2x - 32 = 0
> Solve the following equations for x 32x – 12*3x + 27 = 0
> Solve the following equations for x 22x+2 - 17*2x + 4 = 0
> Solve the following equations for x 22x - 6*2x + 8 = 0
> Solve the following equations for x 2x – 1/2x = 0
> Compute d/dx f (g (x)), where f (x) and g (x) are: f (x) = (x3 + 1)2, g (x) = x2 + 5
> Solve the following equations for x 2x – 8/22x = 0
> Solve the following equations for x (2 - 3x)5x + 4 * 5x = 0
> Write expression in the form 2kx or 3kx, for a suitable constant k. 82x/3, 93x/2, 16-3x/4
> Solve the following equations for x (1 + x)2-x - 5*2-x = 0
> Solve the following equations for x. 35x * 3x - 3 = 0
> Differentiate the functions. y = (5x + 1)(x2 - 1) + (2x + 1)/3
> Solve the following equations for x. 23x = 4 * 25x
> Solve the following equations for x. (32x * 32)4 = 3
> Solve the following equations for x. (2x+1 * 2-3)2 = 2
> Solve the following equations for x. 4(2.7)2x-1 = 10.8
> Compute d/dx f (g (x)), where f (x) and g (x) are: f (x) = 4/x + x2, g (x) = 1 - x4
> Solve the following equations for x. 3(2.7)5x = 8.1
> Solve the following equations for x. 24-x = 8
> Solve the following equations for x. 101-x = 100
> Solve the following equations for x. (3.2)x-3 = (3.2)5
> Write expression in the form 2kx or 3kx, for a suitable constant k. 27x, (3√2)x, (1/8)x
> Solve the following equations for x. (2.5)2x+1 = (2.5)5
> Differentiate the functions. y = [(-2x3 + x)(6x - 3)]4
> Differentiate the following functions. y = 5(√x - 1)4(√x - 2)2
> Differentiate the following functions. y = (2x + 1)5/2(4x - 1)3/2
> Differentiate the following functions. y = x(x5 - 1)3
> Compute d/dx f (g (x)), where f (x) and g (x) are: f (x) = x4 - x2, g (x) = x2 - 4
> Differentiate the following functions. y = 2(5 - x)3(6x - 1)
> Differentiate the following functions. y = (4x - 1)(3x + 1)4
> An investor initially invests $10,000 in a risky venture. Suppose that the investment earns 20% interest, compounded continuously, for 5 years and then 6% interest, compounded continuously, for 5 years thereafter. (a) How much does the $10,000 grow to af
> A stock portfolio increased in value from $100,000 to $117,000 in 2 years. What rate of interest, compounded continuously, did this investment earn?
> Solve the following equations for x. 10-x = 102
> From January 1, 2010, to January 1, 2017, the population of a state grew from 17 million to 19.3 million. (a) Give the formula for the population t years after 2010. (b) If this growth continues, how large will the population be in 2020? (c) In what year
> A piece of charcoal found at Stonehenge contained 63% of the level of 14C found in living trees. Approximately how old is the charcoal?
> The half-life of the radioactive element tritium is 12 years. Find its decay constant.
