1.99
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(a) Approximate f by a Taylor polynomial with degree n at the number a. (b) Graph f and Tn on a common screen. (c) Use Taylorâ€™s Inequality to estimate the accuracy of the approximation / when x lies in the given interval. (d) Check your result in part (c) by graphing /

f(x) = T«(x)

** >** For the limit illustrate Definition 2 by finding values of that correspond to Îµ= 0.2 and Îµ = 0.1.
lim (x – 3x + 4) = 6

** >** Use a graph to find a number δ such that if |x - 1| < δ then | 2x/x2 + 4 - 0.4| < 0.1

** >** Use a graph to find a number δ such that if |x - π/4 | < δ then |tan x - 1| < 0.2

** >** Use the graph of the function f to state the value of each limit, if it exists. If it does not exist, explain why. f(x) = 1 / 1 + e1/y
(a) lim f(x) (b) lim f(x) (c) lim f(x)

** >** The graphs of f and t are given. Use them to evaluate each limit, if it exists. If the limit does not exist, explain why.
(a) lim [f(x) + g(x)] (b) lim [f(x) – g(x)] (c) lim [f(x)g(x)] f(x) (d) lim X3 g(x) X-1 (e) lim [r²f(x)] (f) f(-1) + lim g(x)

** >** Given that find the limits that exist. If the limit does not exist, explain why.
lim f(x) = 4 lim g(x) = -2 lim h(x) = 0 (a) lim [f(x) + 5g(x)] (b) lim [g(x)]³ X2 3f (x) (c) lim f(x) (d) lim 2 g(x) g(x) (e) lim 2 h(x) g(x)h(x) (f) lim 2 f(x)

** >** Show by means of an example that limx → a [f(x)g(x)] may exist even though neither limx → a f(x) nor limx → a g(x) exists.

** >** Show by means of an example that limx→a [f(x) = g(x)] may exist even though neither limx→ a f(x) nor limx → a g(x) exists.

** >** If prove that limxâ†’0 f(x) = 0.
|x² if x is rational if x is irrational f(x) =

** >** If / f(x) x2 = 5, find the following limits.
lim

** >** If / f(x) – 8/x - 1 = 10, find / f(x).

** >** If r is a rational function, use Exercise 57 to show that limx→a r(x) = r(a) for every number a in the domain of r.

** >** If p is a polynomial, show that lim x→a p(x) = p(a).

** >** In the theory of relativity, the Lorentz contraction formula expresses the length L of an object as a function of its velocity v with respect to an observer, where L0 is the length of the object at rest and c is the speed of light. Find limvâ

** >** Sketch the graph of the function and use it to determine the values of a for which limx â†’ a f(x) exists.
|1 + sin x if x <0 S(x) cos x if 0<x<T sin x if x>T

** >** If f(x) = [[x]] + [[2x]], show that limx→2 f(x) exists but is not equal to f(2).

** >** Let f(x) = [[cos x]], -Ï€ â‰¤ x â‰¤ Ï€ . a. Sketch the graph off. b. Evaluate each limit, if it exists. c. For what values of a does limx â†’ a f(x) exist?
(i) lim f(x) (ii)

** >** a. If the symbol [[ ]] denotes the greatest integer function defined in Example 10, evaluate b. If n is an integer, evaluate c. For what values of a does limx â†’ a [[x]] exist?
(i) lim. [x] (ii) lim [x] (iii) lim [x] -2+ -2 X-2.4

** >** Let a. Evaluate each of the following, if it exists. b. Sketch the graph of t.
if x<1 3 g(x) : if x = 1 2 — х? х — 3 if 1<x<2 if x>2 (i) lim g(x) (ii) lim g(x) (iii) g(1) (iv) lim g(x) (v) lim g(x) (vi) lim g(x) 2+

** >** Let Find the value of c so that / exists.
if t<2 Vi+c ift> 2 if t> 2 [4 - 31 B(t)

** >** Let a. Find limxâ†’1- f(x) and limxâ†’1+ f(x). b. Does limxâ†’1 f(x) exist? c. Sketch the graph of f.
(x² + 1 if x<1 l(x – 2)? if x> 1 f(x) =

** >** Let g(x) = x2 + x â€“ 6/|x - 2| a. Find b. Does limx â†’ 2 g(x) exist? c. Sketch the graph of t.
(i) lim g(x) (ii) lim g(x) X→2+

** >** Let g(x) = sgn(sinx). a. Find each of the following limits or explain why it does not exist. b. For which values of a does limx â†’ a g(x) not exist? c. Sketch a graph of g.
(i) lim g(x) (ii) lim g(x) (iii) lim g(x) x0- (iv) lim g

** >** The signum (or sign) function, denoted by sgn, is defined by a. Sketch the graph of this function. b. Find each of the following limits or explain why it does not exist.
-1 if x<0 0 if x= 0 1 if x>0 sgn x = (i) lim sgn x x0+ (ii) lim sgn x X0- (i

** >** Find the limit, if it exists. If the limit does not exist, explain why.
lim X0+ |x|/

** >** Sketch the graph of the function and use it to determine the values of a for which limx â†’ a f(x) exists.
1 +x if x<-1 if -1 <x<1 2 - x if x>1 f(x) = {x

** >** Find the limit, if it exists. If the limit does not exist, explain why.
1 lim |x|) X0-

** >** Find the limit, if it exists. If the limit does not exist, explain why.
2 - |x| lim x→-2 2 + x

** >** Find the limit, if it exists. If the limit does not exist, explain why.
2x lim x-0.5- |2x – x²|

** >** Find the limit, if it exists. If the limit does not exist, explain why.
2x + 12 lim x-6 |x + 6|

** >** Find the limit, if it exists. If the limit does not exist, explain why.
lim (2x + |x – 3|)

** >** Prove that
lim r esin(/3) = 0.

** >** Prove that
lim x* cos 2 0.

** >** If 2x â‰¤ g(x) â‰¤ x4 - x2 + 2 for all x, evaluate
lim g(x).

** >** If 4x - 9 â‰¤ f(x) â‰¤ x2 - 4x + 7 for x â‰¥ 0, find
lim f(x). X4

** >** Use the Squeeze Theorem to show that Illustrate by graphing the functions f, g, and h (in the notation of the Squeeze Theorem) on the same screen.
lim Vx3 + x² sin TT %3D

** >** A patient receives a 150-mg injection of a drug every 4Â hours. The graph shows the amount f(t) of the drug in the blood stream after t hours. Find and explain the significance of these one-sided limits.
lim f(1) lim f(1) +12 and +12 f(

** >** A cardiac monitor is used to measure the heart rate of a patient after surgery. It compiles the number of heartbeats after t minutes. When the data in the table are graphed, the slope of the tangent line represents the heart rate in beats per minute. T

** >** A tank holds 1000 gallons of water, which drains from the bottom of the tank in half an hour. The values in the table show the volume V of water remaining in the tank (in gallons) after t minutes. a. If P is the point (15, 250) on the graph of V, find

** >** Find an equation of the parabola with focus (2, 1) and directrix x = -4.

** >** Find the Maclaurin series for f and its radius of convergence. You may use either the direct method (definition of a Maclaurin series) or known series such as geometric series, binomial series, or the Maclaurin series for ex, sin x, tan-1x, and ln(1 + x)

** >** Determine whether the sequence converges or diverges. If it converges, find the limit.
an Vn + 2

** >** Use the Ratio Test to determine whether the series is convergent or divergent.
3" E (-1)"-1 2"n n-1

** >** Evaluate the integral.
1- tan?x sec?x 2,

** >** Find the Maclaurin series for f and its radius of convergence. You may use either the direct method (definition of a Maclaurin series) or known series such as geometric series, binomial series, or the Maclaurin series for ex, sin x, tan-1x, and ln(1 + x)

** >** Find the foci and vertices and sketch the graph.
бу? + х — 36у + 55 — 0

** >** (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpsonâ€™s Rule with n = 10 to approximate the given integral. Round your answers to six decimal places.
dx J2 In x a>

** >** The force due to gravity on an object with mass m at a height h above the surface of the earth is where R is the radius of the earth and t is the acceleration due to gravity for an object on the surface of the earth. (a) Express F as a series in powers o

** >** Use series to evaluate the following limit.
sin x – x lim .3

** >** (a) Approximate f by a Taylor polynomial with degree n at the number a. (b) Graph f and Tn on a common screen. (c) Use Taylorâ€™s Inequality to estimate the accuracy of the approximation / when x lies in the given interval. (d) Check your

** >** Determine whether the series is convergent or divergent. If it is convergent, find its sum.
E (/7 ) * k-0

** >** Determine whether the sequence converges or diverges. If it converges, find the limit.
an = e

** >** Evaluate the indefinite integral as an infinite series.
cos x - 1 dx

** >** (a) Use the reduction formula in Example 6 to show that (b) Use part (a) and the reduction formula to evaluate
sin 2x S sin'x dx + C 4 2 S sin'x dx.

** >** Find the Maclaurin series for f and its radius of convergence. You may use either the direct method (definition of a Maclaurin series) or known series such as geometric series, binomial series, or the Maclaurin series for ex, sin x, tan-1x, and ln(1 + x)

** >** Find the Maclaurin series for f and its radius of convergence. You may use either the direct method (definition of a Maclaurin series) or known series such as geometric series, binomial series, or the Maclaurin series for ex, sin x, tan-1x, and ln(1 + x)

** >** Find an equation of the ellipse with foci (3, ±2) and major axis with length 8.

** >** Find the Maclaurin series for f and its radius of convergence. You may use either the direct method (definition of a Maclaurin series) or known series such as geometric series, binomial series, or the Maclaurin series for ex, sin x, tan-1x, and ln(1 + x)

** >** (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpsonâ€™s Rule with n = 10 to approximate the given integral. Round your answers to six decimal places.
S Vã cos x dx

** >** Evaluate the indefinite integral as an infinite series.
(x²sin(x²) dx 2

** >** Find the radius of convergence of the series
(2n)! (n!)? -1 8

** >** Find the radius of convergence and interval of convergence of the series.
2"(х — 3)" Σ Vn + 3 n-0

** >** Determine whether the sequence converges or diverges. If it converges, find the limit.
n° an n' - 2n

** >** Find the radius of convergence and interval of convergence of the series.
2"(х — 2)" (п + 2)! n-1

** >** Determine whether the series is convergent or divergent. If it is convergent, find its sum.
n² + 1_ 2n² + 1, Σ 2 In

** >** Find the radius of convergence and interval of convergence of the series.
(x + 2)" Σ n4" n-1

** >** Evaluate the integral.
| sin x sec'x dx

** >** Find the radius of convergence and interval of convergence of the series.
x" E (-1) - n25" n-1

** >** Use the sum of the first eight terms to approximate the sum of the series / .Estimate the error involved in this approximation.

** >** Evaluate the indefinite integral. Illustrate, and check that your answer is reasonable, by graphing both the function and its antiderivative (take C âˆ’ 0).
fx'VI +x² dx

** >** (a) Suppose that the dog in Problem 9 runs twice as fast as the rabbit. Find a differential equation for the path of the dog. Then solve it to find the point where the dog catches the rabbit. (b) Suppose the dog runs half as fast as the rabbit. How close

** >** Evaluate the indefinite integral as an infinite series.
SVī+x³ dx

** >** Determine whether the series is convergent or divergent. If it is convergent, find its sum.
Σ in 100) k-1

** >** Evaluate the integral.
arctan y e dy 1+ y? .2

** >** Determine whether each integral is convergent or divergent. Evaluate those that are convergent.
dx 1 – x?

** >** Determine whether each integral is convergent or divergent. Evaluate those that are convergent.
dx -2

** >** Determine whether each integral is convergent or divergent. Evaluate those that are convergent.
dx 5 - x

** >** Determine whether each integral is convergent or divergent. Evaluate those that are convergent.
Jo x

** >** Determine whether the geometric series is convergent or divergent. If it is convergent, find its
2 + 0.5 + 0.125 + 0.03125 + ...

** >** Determine whether the geometric series is convergent or divergent. If it is convergent, find its
2 + 0.5 + 0.125 + 0.03125 + ...

** >** Determine whether each integral is convergent or divergent. Evaluate those that are convergent.
In x

** >** Determine whether the series is convergent or divergent. If it is convergent, find its sum.
k? Σ k2 – 2k + 5 よ-1

** >** Test the series for convergence or divergence.
2*-13*+1 k-1 k*

** >** Calculate, to four decimal places, the first ten terms of the sequence and use them to plot the graph of the sequence by hand. Does the sequence appear to have a limit? If so, calculate it. If not, explain why.
a, = 1 + (-})"

** >** Calculate, to four decimal places, the first ten terms of the sequence and use them to plot the graph of the sequence by hand. Does the sequence appear to have a limit? If so, calculate it. If not, explain why.
(-1)" а, — 2 +. = 2 n

** >** Calculate, to four decimal places, the first ten terms of the sequence and use them to plot the graph of the sequence by hand. Does the sequence appear to have a limit? If so, calculate it. If not, explain why.
3n an 1 + 6л

** >** Find a formula for the general term an of the sequence, assuming that the pattern of the first few terms continues.
{}, -4.%. 16 25 5> 6> .} 3 4

** >** List the first five terms of the sequence.
Un a — 2, аз — 1, а,н

** >** List the first five terms of the sequence.
а, — 2, ал н %3D 1+ а,

** >** List the first five terms of the sequence.
an aj = 6, ant1 n

** >** Determine whether the series is convergent or divergent. If it is convergent, find its sum.
2 + n 11 – 2n

** >** A dog sees a rabbit running in a straight line across an open field and gives chase. In a rectangular coordinate system (as shown in the figure), assume: (i) The rabbit is at the origin and the dog is at the point sL, 0d at the instant the dog first sees

** >** Evaluate the integral. /

** >** Determine whether the series is convergent or divergent. If it is convergent, find its sum.
1 2 + 81 2 27 243 729

** >** List the first five terms of the sequence.
а, — 1, а,м 1, an+1 5а, — 3

** >** List the first five terms of the sequence.
(-1)"n an п! + 1

** >** (a) What is a convergent sequence? Give two examples. (b) What is a divergent sequence? Give two examples.

** >** Test the series for convergence or divergence.
sin 2n n-1 1+ 2" WI

** >** (a) What is a sequence? (b) What does it mean to say that / (c) What does it mean to say that /

** >** In the figure at the right there are infinitely many circles approaching the vertices of an equilateral triangle, each circle touching other circles and sides of the triangle. If the triangle has sides of length 1, find the total area occupied by the cir