Answer: Show that the following equations have at least one solution in the given

Show that the following equations have at least one solution in the given intervals. a. a cosa 2a2 + 3a 1 = 0, [0.2, 0.31 and [1.2, 1.3] b. (a-2)2 -InA =0, [1,2] and [e, 4] c. 2a cos(2a) -(a - 2)2 = 0. [2, 3] and [3,4] d. a (In A)r = 0, [4,5]

Show that the following equations have at least one solution in the given intervals. a. cosx9, [0,1] b. 6'r -A 2 + 3A-2 = 0, [0.1] c. -3tan(2A) + A = 0, [0,1] d. InA - a 2 + |a 1 = 0, [^,1]

Find intervals containing solutions to the following equations. a. a 2,~x 0 b. 2a cos(2a) (a + I)2 = 0 c. 3a - e x = 0 d. a + I 2 sin(7rA) = 0

Find intervals containing solutions to the following equations. a. x - 3~x = 0 b. 4x2 -ex = 0 c. x 3 - 2x2 - 4x + 2 = 0 d. x 3 + 4.001x2 + 4.002x + 1.101 = 0

Find maxu

Find maXa^^ |/(x)| for the following functions and intervals. a. fix) = 2x/(x2 + 1), [0.2] b. /(x) = x 2V(4 - x), [0.4] c. /(x) = x 3 - 4x + 2, [1,2] d. /(x) = xV(3 - x 2 ), [0.1]

Show that fix) is 0 at least once in the given intervals. a. fix) = 1 - +ie- 1) sin((7r/2)x), [0. 1] b. /(x) = (x 1) tanx 4-x sinTrx, [0,1] c. fix) x sin ttx (x 2) In x, [1,2] d. fix) = (x - 2) sinx ln(x + 2), [-1,3]

Suppose / G C\a, b] and fix) exists on (a, b). Show thatif fix) f 0 for all x in (a, b), then there can exist at most one number p in [a, b] with fip) 0.

Let fix) = x 3 . a. Find the second Taylor polynomial W about xq = 0. b. Find #2(0.5) and the actual error in using #2(0-5) to approximate /(0.5). c. Repeat part (a) using xq = 1. d. Repeat part (b) using the polynomial from part (c).

Find the third Taylor polynomial Pfx) for the function fix) fx + I about xo = 0. Approximate f0l>, f0J5, fL25, and f~L5 using #3(x) and find the actual errors

Find the second Taylor polynomial #20*-') for the function /(x) = e x cosx about xy = 0. a. Use #2(0.5) to approximate /(0.5). Find an upper bound for error |/(0.5) #2(0.5)| using the error formula and compare it to the actual error. b. Find a bound for the error |/(x) #2U')I ' n using #2(x) to approximate fix) on the interval [0. 1]. c. Approximate fix) dx using #2(x) dx. d. Find an upper bound for the error in (c) using f 1 |#2(^) dx\ and compare the bound to the actual error

Repeat Exercise 11 using xy = 7r/6.

Find the third Taylor polynomial Pfx) for the function fix) = (x 1) Inx about xy = 1. a. Use #3(0.5) to approximate /(0.5). Find an upper bound for error \f(0.5) #3(0.5)| using the error formula and compare it to the actual error. b. Find a bound for the error |/(x) #3(x)| in using #3(x) to approximate fix) on the interval [0.5, 1.5]. c. Approximate f0'^ f(x) dx using P^(x) dx. d. Find an upper bound for the error in (c) using J^5 |#3(x) dx\ and compare the bound to the actual error.

Let fix) 2x cos(2x) (x 2)2 and xy = 0. a. Find the third Taylor polynomial #3(x) and use itto approximate /(0.4).b. Use the error formula in Taylor's Theorem to find an upper bound forthe error |/(0.4) P3(0.4)|. Compute the actual error. c. Find the fourth Taylor polynomial P^x) and use it to approximate /(0.4). d. Use the error formula in Taylor's Theorem to find an upper bound forthe error | /(0.4) ^4(0.4)|. Compute the actual error

Find the fourth Taylor polynomial /^(x) for the function f{x) xex ~ about xq 0. a. Find an upper bound for \f(x) ^(x)!, forO < x < 0.4. b. Approximate Jj1 ' 4 /(x) dx using J0 04 P^ix) dx. r0 4 c. Find an upper bound for the error in (b) using j0' /^(x) dx. d. Approximate /'(0.2) using P4'(0.2) and find the error

Use the error term of a Taylor polynomial to estimate the error involved in using sinx ^ x to approximate sin 1

Use a Taylor polynomial about 7r/4 to approximate cos 42 to an accuracy of 10-6

Let /(x) = (1 x)_1 and Xo = 0. Find the nth Taylor polynomial P(x) for /(x) about xq. Find a value ofn necessary for Pn(x) to approximate /(x) to within ID-6 on [0,0.5J

Let /(x) = e x and xo = 0. Find the nth Taylor polynomial P(x) for /(x) about xq. Find a value of n necessary for P(x) to approximate /(x) to within 10-6 on [0, 0.5].

Find the nth Maclaurin polynomial Pn(x) for /(x) = arctanx

The polynomial P2(x) = 1 ^x2 is to be used to approximate/(x) = cosx in Find a bound for the maximum error.

Use the Intermediate Value Theorem 1.11 and Rolle's Theorem 1.7 to show thatthe graph of/(x) = x 3 + 2x + k crosses the x-axis exactly once, regardless of the value ofthe constant k

A Maclaurin polynomial for e x is used to give the approximation 2.5 to e. The error bound in this approximation is established to be P = g. Find a bound for the error in E.

The errorfunction defined by 2 C x erf(x) / e~'' dt -Jk JO gives the probability that any one of a series oftrials will lie within x units ofthe mean, assuming that the trials have a normal distribution with mean 0 and standard deviation n/2/2. This integral cannot be evaluated in terms of elementary functions, so an approximating technique must be used. a. Integrate the Maclaurin series for e~x ' to show that erf(x) * (2k+\)k\ b. The error function can also be expressed in the form 2 2 k x 2k+] erf(x) e > . V* 1 3-5---(2 +1) Verify thatthe two series agree for A = 1, 2, 3, and 4. [Hint: Use the Maclaurin series for e~x '.] c. Use the series in part (a) to approximate erf(l) to within ID-7 . d. Use the same number ofterms as in part (c) to approximate erf(l) with the series in part (b). e. Explain why difficulties occur using the series in part (b) to approximate erf(x).

The nth Taylor polynomial for a function / at xo is sometimes referred to as the polynomial ofdegree at most n that "best" approximates / near xo. a. Explain why this description is accurate.b. Find the quadratic polynomial that best approximates a function / near xq = I if the tangent line at xq = 1 has equation y 4x 1 and if /"(I) = 6.

Prove the Generalized Rolle's Theorem, Theorem 1.10, by verifying the following. a. Use Rolle's Theorem to show that / (z,) = 0 for n 1 numbers in [a, b] with a < z\ < Zi < < Zn-i < b. b. Use Rolle's Theorem to show that / (vv,) 0 for n 2 numbers in [a, h] with 21 < wq < zi < W2 VVn_2 < Zn-I < b. c. Continue the arguments in parts (a) and (b) to show that for each j 1,2,... , n 1, there are n j distinct numbers in [a, b], where / (,) is 0. d. Show that part (c) implies the conclusion ofthe theorem

Example 3 stated that for all x we have | sinx| < |x|. Use the following to verify this statement. a. Show that for all x > 0, /(x) = x sinx is nondecreasing, which implies that sinx < x with equality only when x = 0. b. Use the fact that the sine function is odd to reach the conclusion

A function /: [a, ft] IR is said to satisfy a Lipschitz condition with Lipschitz constant L on [a, ft] if, for every x, y G [a, ft], we have |/(x) /(y)| < L|x y|. a. Show that if / satisfies a Lipschitz condition with Lipschitz constant L on an interval [a, ft], then / C[a, ft]. b. Show that if / has a derivative that is bounded on [a, ft] by L, then / satisfies a Lipschitz condition with Lipschitz constant L on [a, ft]. c. Give an example of a function that is continuous on a closed interval but does not satisfy a Lipschitz condition on the interval.

Suppose / e C[a, ft] and x\ and X2 are in [a, ft]. a. Show that a number if exists between x\ and X2 with /U,) + /(X2) \ c/ _ 1 ^ ^ /(?) = 2 = 2 2* 2)' b. Suppose C| and C2 are positive constants. Show that a number if exists between xi and X2 with C\f{X\) +C2/(X2) /(?) = Ci +C2 c. Give an example to show that the result in part (b) does not necessarily hold when c\ and cq have opposite signs with C| C2.

Let / g C\a, ft], and let p be in the open interval (a, ft). a. Suppose f{p) ^ 0. Show that a 5 > 0 exists with /(x) ^ 0, for all x in [/? 5, /? + 5], with fp (5, p + <5] a subset of [a, ft], b. Suppose /(p) = 0 and ft > 0 is given. Show that a 5 > 0 exists with |/(x)| < ft, for all x in [p (5, p + >5], with [p 5, p + 3] a subset of [a, ft].