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Questions and Answers of Calculus

Calculate, accurate to four decimal places?
Calculate, accurate to five decimal places?
By writing 1/x = 1/[1 - (1 - x)] and using the known expansion of 1 / (1 - x), find the Taylor series for 1 / x in powers of x - 1?
Let f(x) = (1 + x)1/2 + (1 - x)1/2, find the Maclaurin series for f and use it to find f(4) (0) and f(51) (0)?
One can sometimes find a maclaurin series by the method of equating coefficients. For Example, let tan x = sin x / cos x = a0 + a1x + a2x2 + ... x - x3 / 6 + ... = (a0 + a1x +a2x2 + ...) (1 - x2/2 +
Use the method of Problem 34 to find the terms through x4 in the series for sech x?
Use the method of Problem 34 to find the terms through x4 in the series for sech x? Discuss.
Prove Theorem D as follows: Let(a) Show that the series converges for |x| (b) Show that (1 + x)f'(x) = pf(x) and f(0) = 1. (c) Solve this differential equation to get f(x) = (1 + x)p?
Let(a) Show that f'(0) = 0 by using the definition of the derivative. (b) Show that f"(0) = 0. (c) Assuming the known fact that P(n)(0) = 0 for all n, find the Maclaurin series for f(x). (d) Does the
Use a CAS to find the first four nonzero terms in the Maclaurin series for each of the following. Check Problems 1-4 to see that you get the same answers using the methods of Section 9.7? 1. Sin (Exp
In Problem 1-4, find the Maclaurin polynomial of order 4 for f(x) and use it to approximate f(0.12)?1. f(x) = e2x2. f(x) = e-3x3. f(x) = sin 2x4. f(x) = tan x
Find the Taylor polynomial of order 3 based at 1 for f(x) = x3 - 2x2 + 3x + 5 and show that it is an exact representation of f(x)?
Find the Taylor polynomial of order 4 based at 2 for f(x) = x4 and show that it represents f(x) exactly?
Find the Maclaurin polynomial of order n for f(x) = 1 / (1 - x). Then use it with n = 4 so approximate each of the following? (a) f(0.1) (b) f(0.5) (c) f(0.9) (d) f(2)
Find the Maclaurin polynomial of order n (n odd) for sin x. Then use it with n = 5 to approximate each of the following. (This example should convince you that the Maclaurin approximation can be
In Problems 1-4, plot on the same axes the given function along with Maclaurin polynomials of orders 1, 2, 3, and 4? 1. Cos 2x 2. Sin x 3. Sin x2 4. Cos (x - ()
In Problems 1-4, find a good bound for the maximum value of the given expression, given that c is in the stated interval. Answers may vary depending on the technique used?1. |e2c + e-2c|; [0, 3]2.
In Problems 1-4, find a formula for R6(x), the remainder for the Taylor polynomial of order 6 based at a. Then obtain a good bound for |R6(0.5)|. See Examples 4 and 6?1. In(2 + x); a = 02. e-x; a =
Determine the order n of the Maclaurin polynomial for ex that is required to approximate ex to five decimal places, that is, so that | Rn(1) | ( 0.000005?
Determine the order n of the Maclaurin polynomial for 4 tan-1 x that is required to approximate ( = 4 tan-1 to five decimal places, that is, so that |Rn(1)| ( 0.000005?
Find the third order Maclaurin Polynomial for (1 + x)1/2 and bound the error R3(x) for -0.5 ( x ( 0.5?
Find the third order Macularin polynomial for (1 + x)3/2 and bound the error R3(x) if -0.1 ( x ( 0?
Find the third order Maclaurin polynomial for (1 + x)-1/2 and bound the error R3(x) if -0.05 ( x ( 0.05?
Find the fourth order Macluarin polynomial for 1n[(1 + x) / (1 - x)]And bound the error R4(x) for -0.5 ( x ( 0.50?
The fourth order Maclaurin polynomial for sin x is really of third degree since the coefficient of x4 is 0. Thus,Sin x = x - x3 / 6 + R4(x)Show that if 0 ( x ( 0.5, | R4(x) | ( 0.0002605. Use this
In analogy with Problem 49,Cos x = 1 - x2 / 2 + x4 / 24 + R5(x)If 0 ( x ( 1, give a good bound for |R5(x)|. Then use your result to approximateAnd give a bound for the error?
Problem 49 suggests that if n is odd, then the nth order Maclaurin polynomial for sin x is also the (n + 1)st order polynomial, so the error can be calculated using Rn+1. Use this result to find how
Problem 50 suggests that if n is even, then the nth order Maclaurin polynomial for cos x is also the (n + 1)st order polynomial, so the error can be calculated using Rn+1. Use this result to find how
Use a Maclaurin polynomial to obtain the approximation A ( r2t3 / 12 for the area of the shaded region in Figure 5. First express A exactly, then approximate.
If an object of rest mass m0 has velocity v, then (according to the theory of relativity) its mass m is given by m = m0 /(1 - v2 / c2, where c is the velocity of light. Explain how physicists get the
The author of a biology text claimed that the smallest positive solution to x = 1 - e - (1 + k)x is approximately x = 2k, provided k is very small. Show how she reached this conclusion and check it
Expand x4 - 3x3 + 2x2 + x - 2 in a Taylor polynomial of order 4 based at 1 and show that R4(x) = 0 for all x?
Let f(x) be a function that possesses at least n derivatives at x = a and let Pn(x) be the Taylor polynomial of order n based at a. Show that Pn(a) = f(a), P'n(a) = f'(a), P"n(a) = f"(a), ... P(n)n
Calculate sin 43° = sin (43( / 180) by using the Taylor polynomial of order 3 based at (/4 for sin x. Then obtain a good bound for the error made. See Example 6?
Calculate cos 63° by the method illustrated in Example 6. Choose it large enough so that |Rn| ( 0.0005?
Use Maclaurin's Formula, rather than 1'Hpital's Rule to find?a.b.
Let g(x) = p(x) + xn+1f(x), where p(x) is a polynomial of degree at most n and f has derivations through order n Show that p(x) is the Maclaurin polynomial of order n for g?
Recall that the Second Derivative Test for Local Extreme (Section 3.3) does not apply when f"(c) = 0. Prove the following generalization, which may help determine a maximum or a minimum when f"(c) =
In Problems 1-4, find the Taylor polynomial of order 3 based at a for the given function?1. ex; a = 12 Sin x; a = ( / 43. Tan x; a = ( / 62. Sec x; a = ( / 4
For the graph of y = x2 / 4, find the equation of the tangent line and the normal line (i.e., the line perpendicular to the tangent line) that pass through the point (2, 1)?
In Problems 1-2, determine the values of x and y?1.2.
Find all points o the graph of y = x2 / 4 (a) where the tangent line is parallel to the line y = x and (b) where the normal line is parallel to the line y = x?
Find all points of intersection of x2 / 16 + y2 / 9 = 1 and x2 / 9 + y2 / 16 = 1?
Find all point of intersection of x2 / 16 + y2 / 9 = 1 and x2 + y2 = 9?
Use implicit differentiation to find the equation of the tangent line to the curve x2 + y2 / 4 = 1 at the point (- (3 / 2, 1)?
Use implicit differentiation to find the equation of the tangent line to the curve x2 / 9 - y2 / 16 = 1 at the point (9, 8(2)?
Suppose that x = 2 cos t and y = 2 sin t. Fill in the below and plot the ordered pairs (x, y)?
In Problems 1-2, determine the values of r and (?1.2.
In Problems 1-3, plot the curve whose parametric equation is given. 1. x= 2t, y = t - 3; 1 ≤ t ≤ 4 2. x = t/2, y = t2; --1 ≤ t ≤ 2 3. x = 2 cos t, y = 2 sin t; 0 ≤ t ≤ 2π
Find the point on the line y = 2x + 1 that is closest to the point (0, 3), minimum distance between the point and the line?
Find parametric equations of the form x = a1t b1 and y = a2t + b2 through points (1, -1) and (3, 3).
An object moving along the x-axis has position s(t) = t2 - 6t + 8. a. Find the velocity and acceleration. b. When is the object moving forward?
An object initially at rest at position x = 20 has acceleration a = 2. a. Find the velocity and position. b. When will the object reach position 100?
In Problems a-c, sketch a plot of the given conic section. a. 8x = y2 b. x2/4 + y2/9 = 1 c. x2 - 4y2 = 0
In Problems 1 to 3, sketch a graph of the given polar equation. a. r = 2 b. θ = π / 6 c. r = 4 sin θ
In Problems a-c, find expressions for x and y in terms of h and Î¸.a.b.
In Problems a to b, find the length of the given curve. a. x = t, y = 3t3/2, 0 ≤ t ≤ 4 b. x = t + 2, y = 2t - 3; 1 ≤ t ≤ 5
In Problems 1-3, find the coordinates of the focus and the equation of the directrix for each parabola. Make a sketch showing the parabola, its focus, and its directrix. 1. y2 = 4 2. y2 = -12x 3. x2
Find the equation of the parabola with vertex at the origin and axis along the x-axis if the parabola passes through the point (3, -1). Make a sketch.
Find the equation of the parabola through the point (-2, 4) if its vertex is at the origin and its axis is along the x-axis. Make a sketch.
Find the equation of the parabola through the point (6, -5) if its vertex is at the origin and its axis is along the y-axis. Make a sketch.
Find the equation of the parabola whose vertex is the origin and whose axis is the y-axis if the parabola passes through the point (-3, 5). Make a sketch.
In Problems a-c, find the equations of the tangent and the nor-mal lines to the given parabola at the given point. Sketch the parabola, the tangent line, and the normal line a. y2 = 16x, (1, -4) b.
The slope of the tangent line to the parabola y2 = 5x at a certain point on the parabola is \/5/4. Find the coordinates of that point. Make a sketch.
The slope of the tangent line to the parabola x2 = -14y at a certain point on the parabola is -2√7/7. Find the coordinates of that point.
Find the equation of the tangent line to the parabola y2 = -18x that is parallel to the line 3x - 2y + 4 = 0.
Any line segment through the focus of a parabola, with end points on the parabola, is a focal chord. Prove that the tan-gent lines to a parabola at the end points of any focal chord intersect on the
Prove that the tangents to a parabola at the extremities of any focal chord are perpendicular to each other.
A chord of a parabola that is perpendicular to the axis and 1 unit from the vertex has length 1 unit. How far is it from the vertex to the focus?
Prove that the vertex is the point on a parabola closest to the focus.
An asteroid from deep space is sighted from the earth moving on a parabolic path with the earth at the focus. When the line from the earth to the asteroid first makes an angle or 90° with the axis
Work Problem 34, assuming that the angle is 75o rather than 90o.
The cables for the central span of a suspension bridge take the shape of a parabola. If the towers are 800 meters apart and the cables are attached to them at points 400 meters above the floor of the
The focal chord that is perpendicular to the axis of a parabola is called the laths rectum. For the parabola y2 = 4px in Figure 11, let F be the focus, R be any point on the parabola to the left of
For the parabola y2 = 4px in Figure 12. P is any of its points except the vertex, PB is the normal line at P, PA is per-pendicular to the axis of the parabola. and A and B are on the axis. Find |AB|
Show that the focal chord of the parabola y2 = 4px with end points (x1, y1) and (x2, y2) has length x1 + x2 + 2p, Specialize to find the length L of the latus rectum.
Show that the set of points equidistant from a circle and a line outside the circle is a parabola.
Consider a bridge deck weighing S pounds per linear foot and supported by a cable, which is assumed to be of negligi-ble weight compared to the bridge deck. The cable section OP from the lowest point
Consider the parabola y = x2 over the interval [a, b], and let c = (a + b)/2 be the midpoint of [a, b], d be the midpoint of [a, c], and e be the midpoint [c, b]. Let T1 be the triangle with vertices
Illustrate Problems 30 and 31 for the parabola y = 1/4 x2 + 2by plotting (in the same graph window) the parabola, its directrix, its focal chord parallel to the x-axis, and the tan-gent lines at the
In Problem 60 of Section 6.9 you were asked to find the equation of the Gateway Arch in St. Louis, Missouri.a. Find the equation of a parabola with the properties that its vertex is at (0, 630) and
In Problems a-f, find the standard equation of each parabola from the given information. Assume that the vertex is at the origin. a. Focus is at (2, 0) b. Directrix is x = 3 c. Directrix is y - 2 =
In Problem 1-8, name the conic (horizontal ellipse, vertical hyperbola, and so on) corresponding the given equation. 1. x2/9 + y2/4 = 1 2. x2/9 - y2/4 = 1 3. - x2/9 + y2/4 = 1 4. -x2/9 + y2/4 = -1 5.
In Problem 1-3, find the equation of the given central contic. 1. Ellipse with a focus at (-3, 0) and a vertex at (6, 0) 2. Ellipse with a focus at (6, 0) and eccentricity 2/3 3. Ellipse with a focus
In Problem a-c, find the equation of the set of points P satisfying the given conditions. a. The sum of the distances of P from (0, ±9) is 26. b. The sum of the distances of P from (±4, 0) is
In Problem 1-3 find the equation of the tangent line to the given curve at the given point. 1. x2/27 + y2/9 = 1 at (3, √6) 2. x2/24 + y/16 = 1 at (3√2, -2) 3.x2/27 + y2/9 = 1 at (3, - √6)
A doorway in the shape of an elliptical arch (a half-ellipse) is 10 feet wide and 4 feet high at the center. A box 2 feet high is to be pushed through the doorway. How wide can the box be?
How high is the arch of Problem 43 at a distance 2 feet to the right of the center? In Problem 43 A doorway in the shape of an elliptical arch (a half-ellipse) is 10 feet wide and 4 feet high at the
How long is the lams rectum (chord through the focus perpendicular to the major axis) for the ellipse x2/a2 + y2/b2 = 1?
Determine the length of the latus rectum of the hyperbola x2/a2 - y2/b2 = 1.
Halley's comet has an elliptical orbit with major and minor diameters of 36.18 AU and 9.12 AU, respectively (1 AU is 1 astronomical unit, the earth's mean distance from the sun). What is its minimum
The orbit of the comet Kahoutek is an ellipse with eccentricity e = 0.999925 with the sun at a focus. If its minimum distance to the sun is 0.13 AU, what is its maximum distance from the sun?
In 1957, Russia launched Sputnik I. Its elliptical orbit around the earth reached maximum and minimum distances from the earth of 583 miles and 132 miles, respectively. Assuming that the center of
The orbit of the planet Pluto has an eccentricity 0.249. The closest that Pluto comes to the sun is 29.65 AU, and the farthest is 49.31 AU. Find the major and minor diameters.
If two tangent lines to the ellipse 9x2 + 4y2 = 36 inter-sect the y-axis at (0, 6), find the points of tangency.
If the tangent lines to the hyperbola 9x2 - y2 = 36 inter-sect the y-axis at (0, 6), find the points of tangency.
The slope of the tangent line to the hyperbola 2x2 - 7y2 - 35 = 0 at two points on the hyperbola is - 2/3. What are the coordinates of the points of tangency?
Find the equations of the tangent lines to the ellipse x2 + 2y2 - 2 = 0 that are parallel to the line 3x - 3√2y - 7 = 0
Find the area of the ellipse b2x2 + a2y2 = a2b2.
Find the volume of the solid obtained by revolving the ellipse b2x2 + a2y2 = a2b2 about the y-axis.

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