New Semester Started Get 50% OFF Study Help! --h --m --s Claim Now

Question Answers

Textbooks

Find textbooks, questions and answers

Oops, something went wrong!

Change your search query and then try again

Result Not Found

Books FREE
Study Help
Tutors
AI Study Planner NEW
Sell Books

Search
Sign In
Register

study help
mathematics
precalculus

Thomas Calculus Early Transcendentals 13th Edition Joel R Hass, Christopher E Heil, Maurice D Weir - Solutions

Does g(x) = csc x + 2 cot x have any local maximum values? Give reasons for your answer.
Is it true that a discontinuous function cannot have both an absolute maximum and an absolute minimum value on a closed interval? Give reasons for your answer.
How do you find the absolute extrema of a continuous function on a closed interval? Give examples.
Use Newton’s method to estimate the two zeros of the function ƒ(x) = x4 + x - 3. Start with x0 = -1 for the left-hand zero and with x0 = 1 for the zero on the right. Then, in each case, find x2.
Does ƒ(x) = (7 + x)(11 - 3x)1/3 have an absolute minimum value? An absolute maximum? If so, find them or give reasons why they fail to exist. List all critical points of ƒ.
Can you conclude anything about the extreme values of a continuous function on an open interval? On a half-open interval? Give reasons for your answer.
What are the hypotheses and conclusion of Rolle’s Theorem? Are the hypotheses really necessary? Explain.
Use Newton’s method to estimate the two zeros of the function ƒ(x) = 2x - x2 + 1. Start with x0 = 0 for the left-hand zero and with x0 = 2 for the zero on the right. Then, in each case, find x2.
What are the hypotheses and conclusion of the Mean Value Theorem? What physical interpretations might the theorem have?
Use Newton’s method to find the positive fourth root of 2 by solving the equation x4 - 2 = 0. Start with x0 = 1 and find x2.
Does g(x) = ex - x have an absolute minimum value? An absolute maximum? If so, find them or give reasons why they fail to exist. List all critical points of g.
a. Suppose that the first derivative of y = ƒ(x) is y′ = 6(x + 1)(x - 2)2. At what points, if any, does the graph of ƒ have a local maximum, local minimum, or point of inflection?b. Suppose that the first derivative of y = ƒ(x) is y′ = 6x(x + 1)(x - 2). At what points, if any, does the graph
Use Newton’s method to find the negative fourth root of 2 by solving the equation x4 - 2 = 0. Start with x0 = -1 and find x2.
Does ƒ(x) = 2ex/(1 + x2) have an absolute minimum value? An absolute maximum? If so, find them or give reasons why they fail to exist. List all critical points of ƒ.
If ƒ′(x) ≤ 2 for all x, what is the most the values of ƒ can increase on [0, 6]? Give reasons for your answer.
How can you sometimes identify a function ƒ(x) by knowing ƒ' and knowing the value of ƒ at a point x = x0? Give an example.
a. Show that -1/2 ≤ x/(1 + x2) ≤ 1/2 for every value of x.b. Suppose that ƒ is a function whose derivative is ƒ′(x) = x/(1 + x2). Use the result in part (a) to show thatfor any a and b. |v- q² = |(v)ƒ — (9)ƒ|
Suppose that your first guess is lucky, in the sense that x0 is a root of ƒ(x) = 0. Assuming that ƒ′(x0) is defined and not 0, what happens to x1 and later approximations?
Find the absolute maximum and absolute minimum values of ƒ over the interval.ƒ(x) = x - 2 ln x, 1 ≤ x ≤ 3
Suppose that ƒ is continuous on [a, b] and that c is an interior point of the interval. Show that if ƒ′(x) ≤ 0 on [a, c) and ƒ′(x) ≥ 0 on (c, b] , then ƒ(x) is never less than ƒ(c) on [a, b].
What is the First Derivative Test for Local Extreme Values? Give examples of how it is applied.
You plan to estimate π/2 to five decimal places by using Newton’s method to solve the equation cos x = 0. Does it matter what your starting value is? Give reasons for your answer.
Find the absolute maximum and absolute minimum values of ƒ over the interval.ƒ(x) = (4/x) + ln x2, 1 ≤ x ≤ 4
Show that if h > 0, applying Newton’s method toleads to x1 = -h if x0 = h and to x1 = h if x0 = -h. Draw a picture that shows what is going on. f(x) = J√x, x ≥ 0 √=x, x
How do you test a twice-differentiable function to determine where its graph is concave up or concave down? Give examples.
The greatest integer function ƒ(x) = [x], defined for all values of x, assumes a local maximum value of 0 at each point of [0, 1). Could any of these local maximum values also be local minimum values of ƒ? Give reasons for your answer.
The derivative of ƒ(x) = x2 is zero at x = 0, but ƒ is not a constant function. Doesn’t this contradict the corollary of the Mean Value Theorem that says that functions with zero derivatives are constant? Give reasons for your answer.
What is an inflection point? Give an example. What physical significance do inflection points sometimes have?
Points A and B lie at the ends of a diameter of a unit circle and point C lies on the circumference. Is it true that the area of triangle ABC is largest when the triangle is isosceles? How do you know?
a. Give an example of a differentiable function ƒ whose first derivative is zero at some point c even though ƒ has neither a local maximum nor a local minimum at c. b. How is this consistent with Theorem 2? Give reasons for your answer. THEOREM 2-The First Derivative Theorem for Local Extreme
Let h = ƒg be the product of two differentiable functions of x.a. If ƒ and g are positive, with local maxima at x = a, and if ƒ′ and g′ change sign at a, does h have a local maximum at a?b. If the graphs of ƒ and g have inflection points at x = a, does the graph of h have an inflection
Apply Newton’s method to ƒ(x) = x1/3 with x0 = 1 and calculate x1, x2, x3, and x4. Find a formula for |xn| . What happens to |xn| as n→∞? Draw a picture that shows what is going on.
What is the Second Derivative Test for Local Extreme Values? Give examples of how it is applied.
The function y = 1/x does not take on either a maximum or a minimum on the interval 0 < x < 1 even though the function is continuous on this interval. Does this contradict the Extreme Value Theorem for continuous functions? Why?
Use the following information to find the values of a, b, and c in the formula ƒ(x) = (x + a)/ (bx2 + cx + 2).i) The values of a, b, and c are either 0 or 1.ii) The graph of ƒ passes through the point (-1, 0).iii) The line y = 1 is an asymptote of the graph of ƒ.
What do the derivatives of a function tell you about the shape of its graph?
To calculate a planet’s space coordinates, we have to solve equations like x = 1 + 0.5 sin x. Graphing the function ƒ(x) = x - 1 - 0.5 sin x suggests that the function has a root near x = 1.5. Use one application of Newton’s method to improve this estimate. That is, start with x0 = 1.5 and
What are the maximum and minimum values of the function y = |x| on the interval -1 ≤ x < 1? Notice that the interval is not closed. Is this consistent with the Extreme Value Theorem for continuous functions? Why?
For what value or values of the constant k will the curve y = x3 + kx2 + 3x - 4 have exactly one horizontal tangent?
List the steps you would take to graph a polynomial function. Illustrate with an example.
The curve y = tan x crosses the line y = 2x between x = 0 and x = π/2. Use Newton’s method to find where.
A graph that is large enough to show a function’s global behavior may fail to reveal important local features. The graph of ƒ(x) = (x8/8) - (x6/2) - x5 + 5x3 is a case in point.a. Graph ƒ over the interval -2.5 ≤ x ≤ 2.5. Where does the graph appear to have local extreme values or points of
What formula do we need, in addition to the three listed in Question 9, to differentiate rational functions?Question 9Explain how the three formulasa.b.c. d (x”) = nxn-1 dx
The Second Derivative Test for Local Maxima and Minima says:a. ƒ has a local maximum value at x = c if ƒ′(c) = 0 and ƒ″(c) b. ƒ has a local minimum value at x = c if ƒ′(c) = 0 and ƒ″(c) > 0.To prove statement (a), let P = (1/2) |ƒ″(c)|. Then use the fact thatto conclude that for
What is a cusp? Give examples.
Use Newton’s method to find the two real solutions of the equation x4 - 2x3 - x2 - 2x + 2 = 0.
List the steps you would take to graph a rational function. Illustrate with an example.
a. How many solutions does the equation sin 3x = 0.99 - x2 have?b. Use Newton’s method to find them.
a. Show that g(t) = sin2 t - 3t decreases on every interval in its domain.b. How many solutions does the equation sin2 t - 3t = 5 have? Give reasons for your answer.
Describe Newton’s method for solving equations. Give an example. What is the theory behind the method? What are some of the things to watch out for when you use the method?
Explain why the following four statements ask for the same information:i) Find the roots of ƒ(x) = x3 - 3x - 1.ii) Find the x-coordinates of the intersections of the curve y = x3 with the line y = 3x + 1.iii) Find the x-coordinates of the points where the curve y = x3 - 3x crosses the horizontal
You want to bore a hole in the side of the tank shown here at a height that will make the stream of water coming out hit the ground as far from the tank as possible. If you drill the hole near the top, where the pressure is low, the water will exit slowly but spend a relatively long time in the
An American football player wants to kick a field goal with the ball being on a right hash mark. Assume that the goal posts are b feet apart and that the hash mark line is a distance a > 0 feet from the right goal post. (See the accompanying figure.) Find the distance h from the goal post line
Sometimes the solution of a max-min problem depends on the proportions of the shapes involved. As a case in point, suppose that a right circular cylinder of radius r and height h is inscribed in a right circular cone of radius R and height H, as shown here. Find the value of r (in terms of R and H)
Outline a general strategy for solving max-min problems. Give examples.
a. Does cos 3x ever equal x? Give reasons for your answer.b. Use Newton’s method to find where.
a. Show that y = tan θ increases on every open interval in its domain.b. If the conclusion in part (a) is really correct, how do you explain the fact that tan π = 0 is less than tan (π/4) = 1?
Describe l’Hôpital’s Rule. How do you know when to use the rule and when to stop? Give an example.
a. Show that the equation x4 + 2x2 - 2 = 0 has exactly one solution on [0, 1].b. Find the solution to as many decimal places as you can.
Find the four real zeros of the function ƒ(x) = 2x4 - 4x2 + 1.
The graphs of y = x2(x + 1) and y = 1/x (x > 0) intersect at one point x = r. Use Newton’s method to estimate the value of r to four decimal places. 3 نیا 2 1 0 y = x2(x + 1) 1 2 +x
How can you sometimes handle limits that lead to indeterminate forms ∞/∞, ∞ · 0, and ∞ - ∞? Give examples.
In the middle of the fourteenth century, Albert of Saxony (1316–1390) proposed a model of free fall that assumed that the velocity of a falling body was proportional to the distance fallen. It seemed reasonable to think that a body that had fallen 20 ft might be moving twice as fast as a body
You can estimate the value of the reciprocal of a number a without ever dividing by a if you apply Newton’s method to the function ƒ(x) = (1/x) - a. For example, if a = 3, the function involved is ƒ(x) = (1/x) - 3.a. Graph y = (1/x) - 3. Where does the graph cross the x-axis?b. Show that the
Estimate π to as many decimal places as your calculator will display by using Newton’s method to solve the equation tan x = 0 with x0 = 3.
a. Show that ƒ(x) = x/(x + 1) increases on every open interval in its domain.b. Show that ƒ(x) = x3 + 2x has no local maximum or minimum values.
Find the smallest value of the positive constant m that will make mx - 1 + (1/x) greater than or equal to zero for all positive values of x.
How can you sometimes handle limits that lead to indeterminate forms 1∞, 00, and ∞∞? Give examples.
As a result of a heavy rain, the volume of water in a reservoir increased by 1400 acre-ft in 24 hours. Show that at some instant during that period the reservoir’s volume was increasing at a rate in excess of 225,000 gal/min. (An acre-foot is 43,560 ft3, the volume that would cover 1 acre to the
The formula F(x) = 3x + C gives a different function for each value of C. All of these functions, however, have the same derivative with respect to x, namely F′(x) = 3. Are these the only differentiable functions whose derivative is 3? Could there be any others? Give reasons for your answers.
Can a function have more than one antiderivative? If so, how are the antiderivatives related? Explain.
Use the Intermediate Value Theorem from Section 2.5 to show that ƒ(x) = x3 + 2x - 4 has a root between x = 1 and x = 2. Then find the root to five decimal places.Intermediate Value Theorem from Section 2.5 THEOREM 11-The Intermediate Value Theorem for Continuous Functions If f is a continuous
Find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. fox (x + 1) dx
The graph of the position function s = ƒ(t) of an object moving on a coordinate line (t represents time). At approximately what times (if any) is each object’s (a) Velocity equal to zero? (b) Acceleration equal to zero? During approximately what time intervals does the object move (c)
Suppose that it costs a company y = a + bx dollars to produce x units per week. It can sell x units per week at a price of P = c - ex dollars per unit. Each of a, b, c, and e represents a positive constant. (a) What production level maximizes the profit? (b) What is the corresponding
During World War II it was necessary to administer blood tests to large numbers of recruits. There are two standard ways to administer a blood test to N people. In method 1, each person is tested separately. In method 2, the blood samples of x people are pooled and tested as one large sample. If
What is an indefinite integral? How do you evaluate one? What general formulas do you know for finding indefinite integrals?
The graphs of y = √x and y = 3 - x2 intersect at one point x = r. Use Newton’s method to estimate the value of r to four decimal places.
The graph of the position function s = ƒ(t) of an object moving on a coordinate line (t represents time). At approximately what times (if any) is each object’s (a) Velocity equal to zero? (b) Acceleration equal to zero? During approximately what time intervals does the object move (c)
Find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. [ ( 3₁² + 1 ) dt
Calculate the first derivatives of ƒ(x) = x2/(x2 + 1) and g(x) = -1/(x2 + 1). What can you conclude about the graphs of these functions?
How can you sometimes solve a differential equation of the form dy/dx = ƒ(x)?
What is an initial value problem? How do you solve one? Give an example.
In submarine location problems, it is often necessary to find a submarine’s closest point of approach (CPA) to a sonobuoy (sound detector) in the water. Suppose that the submarine travels on the parabolic path y = x2 and that the buoy is located at the point (2, -1/2).a. Show that the value of x
If you know the acceleration of a body moving along a coordinate line as a function of time, what more do you need to know to find the body’s position function? Give an example.
Some curves are so flat that, in practice, Newton’s method stops too far from the root to give a useful estimate. Try Newton’s method on ƒ(x) = (x - 1)40 with a starting value of x0 = 2 to see how close your machine comes to the root x = 1. See the accompanying graph. Slope = -40 1 0 y = (x -
Find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. + 4t³ dt
Find the approximate values of r1 through r4 in the factorization8x4 - 14x3 - 9x2 + 11x - 1 = 8(x - r1)(x - r2)(x - r3)(x - r4).
Graph the curve.y = x2 - (x3/6)
Find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. √(1-x² – 3x³) dx
Find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. (2x³ 5x + 7) dx
Use Newton’s method to find the zeros of ƒ(x) = 4x4 - 4x2 using the given starting values.a. x0 = -2 and x0 = -0.8, lying in (-∞, -(2/2)b. x0 = -0.5 and x0 = 0.25, lying in (-√21/7, √21/7)c. x0 = 0.8 and x0 = 2, lying in (√2/2, ∞)d. x0 = -√21/7 and x0 = √21/7
When a smaller pipe branches off from a larger one in a flow system, we may want it to run off at an angle that is best from some energysaving point of view. We might require, for instance, that energy loss due to friction be minimized along the section AOB shown in the accompanying figure. In this
Assume that the brakes of an automobile produce a constant deceleration of k ft/sec2. (a) Determine what k must be to bring an automobile traveling 60 mi/hr (88 ft/sec) to rest in a distance of 100 ft from the point where the brakes are applied. (b) With the same k, how far would a car
The accompanying figure shows a circle of radius r with a chord of length 2 and an arc s of length 3. Use Newton’s method to solve for r and θ (radians) to four decimal places. Assume 0 < θ < π. 0 2 S = 3
Graph the curve.y = -x3 + 6x2 - 9x + 3
Graph the curve.y = x3 - 3x2 + 3
Find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. / (1₁/²2 - 1x² - 13 ) ₁ dx
Graph the curve y = x√3 - x
Find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation. [(1/3) 2 - 2x) dx +

Showing 11800 - 11900 of 29454

First
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
Last

Step by Step Answers

Services

Sitemap
Fun
Definitions
Become Tutor
Used Textbooks
Study Help Categories
Recent Questions
Expert Questions
Campus Wear
Sell Your Books

Company Info

Security
Copyrights
Privacy Policy
Terms & Conditions
SolutionInn Fee
Scholarship
Online Quiz
Give Feedback, Get Rewards

Get In Touch

About Us
Contact Us
Career
Jobs
FAQ
Student Discount
Campus Ambassador

Secure Payment

Download Our App

© 2026 SolutionInn. All Rights Reserved