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mathematics
calculus

Questions and Answers of Calculus

For what value of does the equation e2x = k √x have exactly one solution?
For which positive numbers is it true that ax > 1 + x for all?
Given an ellipse x2/a2 + y2/b2 = 1, where a ≠ b, find the equation of the set of all points from which there are two tangents to the curve whose slopes are? (a) Reciprocals and (b) Negative
Find the two points on the curve y = x4 – 2x2 – x that have a common tangent line.
Suppose that three points on the parabola y = x2 have the property that their normal lines intersect at a common point. Show that the sum of their -coordinates is 0.
A lattice point in the plane is a point with integer coordinates. Suppose that circles with radius are drawn using all lattice points as centers. Find the smallest value of such that any line with
A cone of radius r centimeters and height h centimeters is lowered point first at a rate of 1 cm/s into a tall cylinder of radius R centimeters that is partially filled with water. How fast is the
Explain the difference between an absolute minimum and a local minimum.
Suppose is a continuous function defined on a closed interval [a, b].(a) What theorem guarantees the existence of an absolute maximum value and an absolute minimum value for f? (b) What steps would
For each of the numbers a, b, c, d, e, r, s, and t, state whether the function whose graph is shown has an absolute maximum or minimum, a local maximum or minimum, or neither a maximum nor a minimum.
Use the graph to state the absolute and local maximum and minimum values of the function.
(a) Sketch the graph of a function that has a local maximum at 2 and is differentiable at 2.(b) Sketch the graph of a function that has a local maximum at 2 and is continuous but not differentiable
(a) Sketch the graph of a function on [–1, 2] that has an absolute maximum but no local maximum.(b) Sketch the graph of a function on [–1, 2] that has a local maximum but no absolute maximum.
(a) Sketch the graph of a function on [–1, 2] that has an absolute maximum but no absolute minimum.(b) Sketch the graph of a function on [–1, 2] that is discontinuous but has both an absolute
(a) Sketch the graph of a function that has two local maxima, one local minimum, and no absolute minimum.(b) Sketch the graph of a function that has three local minima, two local maxima, and seven
Sketch the graph of f by hand and use your sketch to find the absolute and local maximum and minimum values of f. (Use the graphs and transformations of Sections 1.2 and 1.3.)
Find the absolute maximum and absolute minimum values of f on the given interval.
If and are positive numbers, find the maximum value of f(x) = xa (1 – x)b 0 < x < 1.
Use a graph to estimate the critical numbers of f(x) = | x3 – 3x2 + 2 | correct to one decimal place.
(a) Use a graph to estimate the absolute maximum and minimum values of the function to two decimal places.(b) Use calculus to find the exact maximum and minimum values.
Between 0oC and 30oC, the volume V (in cubic centimeters) of 1 kg of water at a temperature T is given approximately by the V = 999.87 – 0.06426T + 0.0085043T2 – 0.0000679T3 formula Find the
An object with weight W is dragged along a horizontal plane by a force acting along a rope attached to the object. If the rope makes an angle with the plane, then the magnitude of the force iswhere
A model for the food-price index (the price of a representative €œbasket€ of foods) between 1984 and 1994 is given by the function where t is measured in years since midyear
On May 7, 1992, the space shuttle Endeavour was launched on mission STS-49, the purpose of which was to install a new perigee kick motor in an Intelsat communications satellite. The table gives the
When a foreign object lodged in the trachea (windpipe) forces v a person to cough, the diaphragm thrusts upward causing an increase in pressure in the lungs, this is accompanied by a contraction of
Show that 5 is a critical number of the function g(x) = 2 + (x – 5)3 but does not have a local extreme value at 5.
Prove that the function f(x) = x101 + x51 + x + 1 has neither a local maximum nor a local minimum.
If f has a minimum value at c, show that the function g(x) = – f (x) has a maximum value at .
Prove Fermat’s Theorem for the case in which f has a local minimum at c.
A cubic function is a polynomial of degree 3; that is, it has the form f(x) = ax3 + bx2 + cx + d, where a ≠ 0. (a) Show that a cubic function can have two, one, or no critical number(s). Give
Verify that the function satisfies the three hypotheses of Role€™s Theorem on the given interval. Then find all numbers that satisfy the conclusion of Role€™s Theorem.
Let f(x) = 1 – x2/3. Show that f(– 1) = f(1) but there is no number c in (–1, 1) such that f’(c) = 0. Why does this not contradict Role’s Theorem?
Let f(x) = (x – 1)–2. Show that f (0) = f (2) but there is no number in (0, 2) such that f’(c) = 0. Why does this not contradict Role’s Theorem?
Use the graph of f to estimate the values of that satisfy the conclusion of the Mean Value Theorem for the interval [0, 8].
Use the graph of f given in Exercise 7 to estimate the values of c that satisfy the conclusion of the Mean Value Theorem for the interval [1, 7].
(a) Graph the function f(x) = x + 4/x in the viewing rectangle [0, 10] by [0, 10].(b) Graph the secant line that passes through the points (1, 5) and (8, 8.5) on the same screen with f.(c) Find the
(a) In the viewing rectangle [–3, 3] by [–5, 5], graph the function f(x) = x2 – 2x and its secant line through the points (– 2, – 4) and (2, 4). Use the graph to estimate the -coordinates
Verify that the function satisfies the hypotheses of the Mean Value Theorem on the given interval. Then find all numbers c that satisfy the conclusion of the Mean Value Theorem.
Let f(x) = | x – 1|. Show that there is no value of such that f(3) –f(0) = f’(c)(3 – 0). Why does this not contradict the Mean Value Theorem?
Let f(x) =(x + 1)/(x – 1). Show that there is nocontradict f(2) – f(0) = f’(c) (2 – 0). Why does this not contradict the Mean Value Theorem?
Show that the equation 1 + 2x + x3 + 4x5 = 0 has exactly one real root.
Show that the equation 2x – 1 – sin x = 0 has exactly one real root.
Show that the equation x3 – 15x + c = 0 has at most one root in the interval [– 2, 2].
Show that the equation x4 + 4x + c = 0 has at most two real roots.
(a) Show that a polynomial of degree 3 has at most three real roots.(b) Show that a polynomial of degree has at most real roots.
(a) Suppose that f is differentiable on R and has two roots. Show that has at least one root.(b) Suppose f is twice differentiable on R and has three roots. Show that has at least one real root.(c)
If f (1) = 10 and f’(x) > 2 for 1 < x < 4, how small can f (4) possibly be?
Suppose that 3 < f’(x) < 5 for all values of x. Show that 18 < f(*) – f(2) < 30.
Does there exist a function f such that f (0) = –1, f (2) = 4, and f’(x) < 2 for all x?
Suppose that f and are continuous [a, b] on and differentiable on (a, b). Suppose also that f(a) = g(a) and f’(x) < g’(x) for a < x < b. Prove that f(b) < g(b) 2.
Show that √1 + x < 1 + ½ x if x > 0.
Suppose f is an odd function and is differentiable everywhere. Prove that for every positive number, there exists a number c in (–b, b) such that f’(c) = f(b)/b.
Use the Mean Value Theorem to prove the inequality |sin a – sin b| < | a – b | for all a and b.
If f’(x) = c (c a constant) for all x, use Corollary 7 to show that f(x) cx + d for some constant d.
Let and f(x) = 1/x andShow that f€™(x) = g€™(x) for all x in their domains. Can we conclude from Corollary 7 that f €“ g is constant?
Use the method of Example 6 to prove the identity 2 sin–1 x = cos –1(1 – 2x2) x > 0
Prove the identity
At 2:00 P.M. a car’s speedometer reads 30 mi/h. At 2:10 P.M. it reads 50 mi/h. Show that at some time between 2:00 and 2:10 the acceleration is exactly 120 mi/h2.
Two runners start a race at the same time and finish in a tie. Prove that at some time during the race they have the same speed.
A number a is called a fixed point of a function f if f (a) = a. Prove that if f’(x) ≠ 1 for all real numbers x, then has f at most one fixed point.
Use the given graph of f to find the following.(a) The largest open intervals on which f is increasing.(b) The largest open intervals on which f is decreasing.(c) The largest open intervals on which
Suppose you are given a formula for a function f.(a) How do you determine where f is increasing or decreasing?(b) How do you determine where the graph of f is concave upward or concave downward?(c)
(a) State the First Derivative Test.(b) State the Second Derivative Test. Under what circumstances is it inconclusive? What do you do if it fails?
The graph of the derivative f€™ of a function f is shown.(a) On what intervals is f increasing or decreasing?(b) At what values of x does have a local maximum or minimum?
The graph of the second derivative f€™€™ of a function f is shown. State the -coordinates of the inflection points of f. Give reasons for your answers.
The graph of the first derivative f €˜of a function f is shown(a) On what intervals is f increasing? Explain.(b) At what values of does f have a local maximum or minimum? Explain.(c) On
Sketch the graph of a function whose first and second derivatives are always negative.
A graph of a population of yeast cells in a new laboratory culture as a function of time is shown.(a) Describe how the rate of population increase varies.(b) When is this rate highest?(c) On what
(a) Find the intervals on which f is increasing or decreasing.(b) Find the local maximum and minimum values of f.(c) Find the intervals of concavity and the inflection points.
Find the local maximum and minimum values of f using both the First and Second Derivative Tests. Which method do you prefer?
(a) Find the critical numbers of f(x) = x4(x – 1)3.(b) What does the Second Derivative Test tell you about the behavior f of at these critical numbers?(c) What does the First Derivative Test tell
Suppose f’’ is continuous on (– ∞, ∞). (a) If f’ (2) = 0 and f’’ (2) = –5, what can you say about f’’? (b) If f’ (6) = 0 and f’’ (6) = 0, what can you say about
Sketch the graph of a function that satisfies all of the given conditions. 0 if x 3. f"(x) < 0 if 1
The graph of the derivative f €˜of a continuous function f is shown(a) On what intervals is f increasing or decreasing?(b) At what values of x does f have a local maximum or minimum?(c) On
(a) Find the intervals of increase or decrease.(b) Find the local maximum and minimum values.(c) Find the intervals of concavity and the inflection points.(d) Use the information from parts
(a) Find the vertical and horizontal asymptotes.(b) Find the intervals of increase or decrease.(c) Find the local maximum and minimum values.(d) Find the intervals of concavity and the inflection
(a) Use a graph of f to estimate the maximum and minimum values. Then find the exact values.(b) Estimate the value of at which f increases most rapidly. Then find the exact value.
(a) Use a graph of f to give a rough estimate of the intervals of concavity and the coordinates of the points of inflection.(b) Use a graph of f€™€™ to give better estimates.
Estimate the intervals of concavity to one decimal place by using a computer algebra system to compute and graph f€™€™.
Let K (t) be a measure of the knowledge you gain by studying for a test for t hours. Which do you think is larger, K (8) - K (7) or K (3) - K (2)? Is the graph of K concave upward or concave
Coffee is being poured into the mug shown in the figure at a constant rate (measured in volume per unit time). Sketch a rough graph of the depth of the coffee in the mug as a function of time.
For the period from 1980 to 2000, the percentage of households in the United States with at least one VCR has been modeled by the functionWhere the time t is measured in years since midyear 1980, so 0
The family of bell-shaped curves occurs in probability and statistics, where it is called the normal density function.The constant μ is called the mean and the positive constant σ is called
Find a cubic function f(x) = ax3 + bx2 + cx + d that has a local maximum value of at –2 and a local minimum value of 0 at 1.
For what values of the numbers and does the function f(x) = axebx2 have the maximum value f(2) = 1?
Suppose f is differentiable on an interval l and f ‘(x) > 0 for all numbers in except for a single number c. Prove that f is increasing on the entire interval l.
Assume that all of the functions are twice differentiable and the second derivatives are never 0.
Show that tan x > x for 0 < x
(a) Show that ex > 1 + x for x > 0.(b) Deduce that ex > 1 + x ½ x2 for x > 0.(c) Use mathematical induction to prove that for x > 0 and any positive integer n,
Show that a cubic function (a third-degree polynomial) always has exactly one point of inflection. If its graph has three x-intercepts x1, x2 and x3, show that the -coordinate of the inflection point
For what values of does the polynomial P(x) = x4 + cx3 + x2 have two inflection points? One inflection points none? Illustrate by graphing P for several values of c. How does the graph change as
Prove that if (c, f(c)) is a point of inflection of the graph of and f’’ exists in an open interval that contains, then f’’(c) = 0.
Show that if f(x) = x4, then f’’ (0) = 0, but (0, 0) is not an inflection point of the graph of f.
Show that the function g(x) = x |x| has an inflection point at (0, 0) but g’’ (0) does not exist.
Suppose that f’’’ is continuous and f’(c) = f’’(c) = 0, but f’’(c) > 0. Does f have a local maximum or minimum at c? Does f have a point of inflection at c?
Given that which of the following limits are indeterminate forms? For those that are not an indeterminate form, evaluate the limit where possible.
Use a graph to estimate the value of the limit. Then use l€™ Hospital€™s Rule to find the exact value.
Illustrate l€™ Hospital€™s Rule by graphing both f(x)/g(x) and f€™(x)/g€™(x) near x = 0 to see that these ratios have the same limit as x → 0. Also
If an initial amount A0 of money is invested at an interest rate i compounded times a year, the value of the investment after t years is
If an object with mass m is dropped from rest, one model for its speed after seconds, taking air resistance into account, is where is the acceleration due to gravity and is a positive constant.(In

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