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mathematics
calculus 4th

Questions and Answers of Calculus 4th

Investigate the behavior and sketch the graphs of y = x2/3 and y = x4/3.
Approximate to three decimal places using Newton’s Method and compare with the value from a calculator.3−1/4
Use the Linear Approximation. A bond that pays $10,000 in 6 years is offered for sale at a price P. The percentage yield Y of the bond isVerify that if P = $7500, then Y = 4.91%. Estimate the drop in
Plot the derivative of ƒ(x) = 3x5 − 5x3. Describe its sign changes and use this to determine the local extreme values of ƒ. Then graph ƒ to confirm your conclusions.
Approximate the largest positive root of ƒ(x) = x4 − 6x2 + x + 5 to within an error of at most 10−4. Refer to Figure 5. 2 3
Find the transition points, intervals of increase/decrease, concavity, and asymptotic behavior. Then sketch the graph, with this information indicated.y = x3 + 24x2
Determine the intervals on which ƒ'(x) is positive and negative, assuming that Figure 15 is the graph of ƒ. निजी 1 2 3 4 5 6 y →X
Determine the intervals on which the function is concave up or down and find the points of inflection. y = 1 2 x² + 3
Use the Linear Approximation. When a bus pass from Albuquerque to Los Alamos is priced at p dollars, a bus company takes in a monthly revenue of R(p) = 1.5p − 0.01p2 (in thousands of dollars).(a)
Find the transition points, intervals of increase/decrease, concavity, and asymptotic behavior. Then sketch the graph, with this information indicated.y = x3 − 3x + 5
Approximate the value specified to three decimal places using Newton’s Method. Use a plot to choose an initial guess.Largest positive root of ƒ(x) = x3 − 5x + 1
Determine the intervals on which ƒ is increasing or decreasing, assuming that Figure 15 is the graph of ƒ'. 2 3 4 5 →X 6
Determine the intervals on which the function is concave up or down and find the points of inflection. y = X x² +9 ୯
Use the Intermediate Value Theorem to prove that sin x − cos x = 3x has a solution, and use Rolle’s Theorem to show that this solution is unique.Rolle's Theorem a f'(c)=0 C f(a)=f(b) y f(c) b X
Use the Linear Approximation.Show that √a2 + b ≈ a + b/2a if b is small. Use this to estimate √26 and find the error using a calculator.
Approximate the value specified to three decimal places using Newton’s Method. Use a plot to choose an initial guess.Negative root of ƒ(x) = x5 − 20x + 10
Approximate the value specified to three decimal places using Newton’s Method. Use a plot to choose an initial guess.Positive solution of sin θ = 0.8θ
Show that ƒ(x) = 2x3 + 2x + sin x + 1 has precisely one real root.
Approximate the value specified to three decimal places using Newton’s Method. Use a plot to choose an initial guess.Positive solution of 4 cos x = x2.
Let x1, x2 be the estimates to a root obtained by applying Newton’s Method with x0 = 1 to the function graphed in Figure 8. Estimate the numerical values of x1 and x2, and draw the tangent lines
Verify the MVT for ƒ(x) = x + 1/x on [2, 5].
Suppose that ƒ(1) = 5 and ƒ'(x) ≥ 2 for x ≥ 1. Use the MVT to show that ƒ(8) ≥ 19.
Find the smallest positive value of x at which y = x and y = tan x intersect.
Use the MVT to prove that if ƒ'(x) ≤ 2 for x > 0 and ƒ(0) = 4, then ƒ(x) ≤ 2x + 4 for all x ≥ 0.
In 1535, the mathematician Antonio Fior challenged his rival Niccolo Tartaglia to solve this problem: A tree stands 12 braccia high; it is broken into two parts at such a point that the height of the
A function ƒ has derivative ƒ(x) = 1/x4 + 1. Where on the interval [1, 4] does ƒ take on its maximum value?
Find (to two decimal places) the coordinates of the point P in Figure 9 where the tangent line to y = cos x passes through the origin. 1 P y = cos x FIGURE 9 2π
In r denotes a yearly interest rate expressed as a decimal (rather than as a percent). If P dollars are deposited every month in an account earning interest at the yearly rate r, then the value S of
Find the critical points and determine whether they are minima, maxima, or neither.ƒ(x) = x3 − 4x2 + 4x
Repeat Exercise 22 but assume that Figure 18 is the graph of the second derivative ƒ". a b nelm c d е If g
Find the critical points and determine whether they are minima, maxima, or neither.s(t) = t4 − 8t2
Find the coordinates of the point on the graph of y = x + 2x−1 closest to the origin in the region x > 0 (Figure 16). 8 6 4 2 y + 1 + 2 y = x + 2x-1 3 X
Find the critical points and determine whether they are minima, maxima, or neither.ƒ(x) = x2(x + 2)3
Find the critical points and determine whether they are minima, maxima, or neither.ƒ(x) = x2/3(1 − x)
If you deposit P dollars in a retirement fund every year for N years with the intention of then withdrawing Q dollars per year for M years, you must earn interest at a rate r satisfyingAssume that
There is no simple formula for the position at time t of a planet P in its orbit (an ellipse) around the sun. Introduce the auxiliary circle and angle θ in Figure 10 (note that P determines θ
Find the critical points and determine whether they are minima, maxima, or neither.g(θ) = sin2 θ + θ
The roots of ƒ(x) = 1/3x3 − 4x + 1 to three decimal places are −3.583, 0.251, and 3.332 (Figure 11). Determine the root to which Newton’s Method converges for the initial choices x0 = 1.85,
Find the critical points and determine whether they are minima, maxima, or neither.h(θ) = 2 cos 2θ + cos 4θ
What happens when you apply Newton’s Method to find a zero of ƒ(x) = x1/3? Note that x = 0 is the only zero.
What happens when you apply Newton’s Method to the equation x3 − 20x = 0 with the unlucky initial guess x0 = 2?
Consider a metal rod of length L fastened at both ends. If you cut the rod and weld on an additional segment of length m, leaving the ends fixed, the rod will bow up into a circular arc of radius R
Consider a metal rod of length L fastened at both ends. If you cut the rod and weld on an additional segment of length m, leaving the ends fixed, the rod will bow up into a circular arc of radius R
Newton’s Method can be used to compute reciprocals without performing division. Let c > 0 and set ƒ(x) = x−1 − c.(a) Show that x − (ƒ(x)/ ƒ(x)) = 2x − cx2.(b) Calculate the first three
Find the extreme values on the interval.ƒ(x) = x2/3 − 2x1/3, [−1, 3]
Find the extreme values on the interval.ƒ(x) = 4x − tan2 x, [−π/4, π/3]
Match the description of ƒ with the graph of its derivative ƒ' in Figure 1.(a) ƒ is increasing and concave up.(b) ƒ is decreasing and concave up.(c) ƒ is increasing and concave down.
Find the critical points and extreme values of ƒ(x) = |x − 1| + |2x − 6| in [0, 8].
Find the points of inflection.y = x3 − 4x2 + 4x
Find the points of inflection.y = x − 2 cos x
Sketch the graph, noting the transition points and asymptotic behavior.y = 12x − 3x2
Sketch the graph, noting the transition points and asymptotic behavior.y = 8x2 − x4
Sketch the graph, noting the transition points and asymptotic behavior.y = x3 − 2x2 + 3
Sketch the graph, noting the transition points and asymptotic behavior.y = 4x − x3/2
Suppose, in the previous exercise, that the warehouse consists of n separate spaces of equal size. Find a formula in terms of n for the maximum possible area of the warehouse.Data from previous
Sketch the graph, noting the transition points and asymptotic behavior. y = X x3 X- x³ + 1
Sketch the graph, noting the transition points and asymptotic behavior. y = X (x² - 4)2/3
Sketch the graph, noting the transition points and asymptotic behavior. y = 1 x + 2 + 1
Sketch the graph, noting the transition points and asymptotic behavior.y = √2 − x3
Sketch the graph, noting the transition points and asymptotic behavior.y = √3 sin x − cos x on [0, 2π]
Draw a curve y = ƒ(x) for which ƒ' and ƒ" have signs as indicated in Figure 2. + -2 0 1 + 3 ++ 5 + X
Sketch the graph, noting the transition points and asymptotic behavior.y = 2x − tan x on [0, 2π]
Find the dimensions of a cylindrical can with a bottom but no top of volume 4 m3 that uses the least amount of metal.
Water is pumped into a sphere at a constant rate (Figure 20). Let h(t) be the water level at time t. Sketch the graph of h (approximately, but with the correct concavity). Where does the point of
A rectangular open-topped box of height h with a square base of side b has volume V = 4 m3. Two of the side faces are made of material costing $40/m2. The remaining sides cost $20/m2. Which values of
The corn yield on a certain farm is Y = −0.118x2 + 8.5x + 12.9 (bushels per acre) where x is the number of corn plants per acre (in thousands). Assume that corn seed costs $1.25 (per thousand
Draw the graph of a function for which ƒ' and ƒ" take on the given sign combinations in order.−+, −−, −+
The function ƒ(x) = x2 + 1/x is concave down for x < 0 and concave up for x > 0. Is there an inflection point at x = 0? Explain.
Apply Newton’s Method to ƒ and initial guess x0 to calculate x1, x2, x3.ƒ(x) = 1 − x sin x, x0 = 7
Compose the absolute value with a familiar function to define a function ƒ that• Has infinitely many local maxima, all of which occur where ƒ = 0, and• Has infinitely many local minima, all of
Estimate using the Linear Approximation or linearization, and use a calculator to estimate the error.5√33
Find a point c satisfying the conclusion of the MVT for the given function and interval.y = (x − 1)(x − 3), [1, 3]
Determine the intervals on which the function is concave up or down and find the points of inflection.y = t3 − 6t2 + 4
Draw the graph of a function for which ƒ' and ƒ" take on the given sign combinations in order.−+, ++, +−
Can a function have an inflection point at a critical point? Explain.
Are the following statements true or false? Explain.(a) FTC I is valid only for positive functions.(b) To use FTC I, you have to choose the right antiderivative.(c) If you cannot find an
Find the general antiderivative of ƒ and check your answer by differentiating.ƒ(x) = 2x4 − 24x2
Which of the following pairs of sums are not equal? (a) (C) - 1) (b) 关 5 (d) i(i + 1),j-1)j j=2
A rainstorm hit Portland, Maine, in October 1996, resulting in record rainfall. The rainfall rate R(t) on October 21 is recorded, in centimeters per hour, in the following table, where t is the
Suppose that F(x) = ƒ(x) and G'(x) = g(x). Which of the following statements are true? Explain.(a) If ƒ = g, then F = G.(b) If F and G differ by a constant, then ƒ = g.(c) If ƒ and g differ by a
Find the maximum volume of a right-circular cone placed upside-down in a right-circular cone of radius R = 3 and height H = 4 as in Figure 3. A cone of radius r and height h has volume 1/3 πr2 h.
A truck gets 10 miles per gallon (mpg) of diesel fuel traveling along an interstate highway at 50 mph. This mileage decreases by 0.15 mpg for each mile per hour increase above 50 mph.(a) If the truck
Redo Exercise 61 for arbitrary R and H.Data From Exercise 61 Find the maximum volume of a right-circular cone placed upside-down in a right-circular cone of radius R = 3 and height H = 4 as in Figure
Show that the maximum area of a parallelogram ADEF that is inscribed in a triangle ABC, as in Figure 4, is equal to one-half the area of ΔABC. A D F B E C
A box of volume 8 m3 with a square top and bottom is constructed out of two types of metal. The metal for the top and bottom costs $50/m2 and the metal for the sides costs $30/m2. Find the
Let ƒ be a function whose graph does not pass through the x-axis and let Q = (a, 0). Let P = (x0,ƒ(x0)) be the point on the graph closest to Q (Figure 5). Prove that PQ is perpendicular to the
Use Exercises 64 and 65 to prove the following assertions for all x ≥ 0 (each assertion follows from the previous one):Data From Exercise 64Prove that if ƒ(0) = g(0) and ƒ'(x) ≤ g'(x) for x ≥
Take a circular piece of paper of radius R, remove a sector of angle θ (Figure 6), and fold the remaining piece into a cone-shaped cup. Which angle θ produces the cup of largest volume? R
Use Newton’s Method to estimate 3√25 to four decimal places.
Use Newton’s Method to find a root of ƒ(x) = x2 − x − 1 to four decimal places.
Three towns A, B, and C are to be joined by an underground fiber cable as illustrated in Figure 39(A). Assume that C is located directly below the midpoint of AB. Find the junction point P that
Tom and Ali drive along a highway represented by the graph of ƒ in Figure 40. During the trip, Ali views a billboard represented by the segment BC along the y-axis. Let Q be the y-intercept of the
A jewelry designer plans to incorporate a component made of gold in the shape of a frustum of a cone of height 1 cm and fixed lower radius r (Figure 43). The upper radius x can take on any value
Compute an area function A(x) of ƒ(x) with lower limit a. Then, to verify the FTC II inverse relationship, compute A (x) and show that it equals ƒ(x). f(x)=4-2x, a = 0
A hot metal object is submerged in cold water. The rate at which the object cools (in degrees per minute) is a function ƒ(t) of time. Which quantity is represented by the integral S f(t) dt?
Which of the following integrals is a candidate for the Substitution Method? (a) (c) S 5x² Sas 5x4 sin(x) dx x³ sin x dx (b) » f sin ³x sin³ x cos x dx
Suppose that F'(x) = ƒ(x) and F(0) = 3, F(2) = 7.(a) What is the area under y = ƒ(x) over [0, 2] if ƒ(x) ≥ 0?(b) What is the graphical interpretation of F(2) − F(0) if ƒ(x) takes on both
Find an antiderivative of the function ƒ(x) = 0.
Water flows into an empty reservoir at a rate of 3000 + 20t L per hour (t is in hours). What is the quantity of water in the reservoir after 5 h?
Find the general antiderivative of ƒ and check your answer by differentiating.ƒ(x) = 18x2

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