Math 231E. Fall 2013. Midterm Practice Problems.Problem 1. Compute the derivatives of the following functionsa. f(x) = sin(x2)exb. f(x) = arcsec(x)c. f(x) = ln(1 + x2+ x4)Problem 2. Give the Taylor polynomial for the following functions to the order indicatedabout the point a = 0a. f(x) = (sin(x) − x)ex, order = 4b. f(x) = (cos(x))/(1 − x), order = 3c. f(x) =11 + x2+ x4, order = 8d. f(x) = x(cos(x2) − 1) sin(x), order = 10e. f(x) = ln(1 + x2), order = 6Problem 3. Give the Taylor series for the following functions to third order about the pointindicateda. f(x) = cos(x), about the point a =π2b. f(x) = sin(x), about the point a =π4c. f(x) =√x, about the point a = 2Problem 4. Some triangle problems!a. Suppose u = cos(x). Express sin(x) in terms of u.b. Given that u = tan(x), express sin(x) in terms of u.Problem 5. Consider the polynomial P (x) = 6 −14x −3x2+ 5x3. Let A, B be the intervalsdefined as:A = {x : −1 < x < 0}, B = {x : 0 < x < 1}Using the Intermediate Value Theorem, in which intervals is the polynomial P (x) guaranteedto have a zero? Explain why.a. Both A and Bb. Only Ac. Only Bd. Neither A nor Be. Not enough informa-tion to decide.Problem 6. Evaluate the following limitsa. limx→11 − 3x2+ 2x3cos(x)b. limx→0|x|cos(x)c. limx→0sin(x)xd. limx→0cos(x2) − 1x4e. limx→0ex3− 1x − sin(x)f. limx→11 + x + x2(x − 1)2g. limx→2sin(πx)ex−2− 1h. limx→∞sin(x2) + 11xx2+ exi. limx→∞x14+ 11x7− 9727x14+ 19x13− 293x5− 5j. limx→0|x|xk. limx→0x2ecos(x)sin(x)Problem 7.f(x) = “the seventh digit after the decimal point in the decimal expansion for x”.DefineA = limx→1−f(x), B = limx→1+f(x).Compute A,
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