Question: Compute dy/dx using the chain rule

> 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?

> 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

> 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.

> One thousand dollars is deposited in a savings account at 10% interest compounded continuously. How many years are required for the balance in the account to reach $3000?

> Find the present value of $10,000 payable at the end of 5 years if money can be invested at 12% with interest compounded continuously.

> Compute d/dx f (g (x)), where f (x) and g (x) are: f (x) = 1/(1 + √x), g (x) = 1/x

> When a rod of molten steel with a temperature of 1800ËšF is placed in a large vat of water at temperature 60F, the temperature of the rod after t seconds is f (t) = 60(1 + 29e-0.15t)ËšF. The graph of this function is shown in Fig. 2.

> The growth of the yellow nutsedge weed is described by a logistic growth formula f (t) of type (9) in Section 5.4. A typical weed has length 8 centimeters after 9 days and length 48 centimeters after 25 days and reaches length 55 centimeters at maturity.

> Refer to Check Your Understanding 5.4. Out of 100 doctors in Group A, none knew about the drug at time t = 0, but 66 of them were familiar with the drug after 13 months. Find the formula for f (t). Check Your Understanding 5.4: A sociological study was

> Consider a demand function of the form q = ae-bp, where a and b are positive numbers. Find E(p), and show that the elasticity equals 1 when p = 1/b.

> A company can sell q = 1000p2e-0.02(p+5) calculators at a price of p dollars per calculator. The current price is $200. If the price is decreased, will the revenue increase or decrease?

> Solve the following equations for x. 52x = 52

> Find the percentage rate of change of the function f (p) = 1/(3p + 1) at p = 1.