Math 231E. Fall 2013. Midterm 2 Practice Problems.Problem 1. Compute the derivatives of the following functionsa. f(x) = cos(x2)esin(x)b. f(x) = arctan(x)c. f(x) = ln(ex(1 + x2+ x3))Problem 2. Consider the polynomial P (x) = 8x3− 36x2+ 46x − 15. We want to find theroots of this polynomial. Find “brackets” for all of the roots of this polynomial of the form[k, k + 1], i.e. find three integers k1, k2, k3such that you can show there is a root r withk1< r < k1+ 1. Use any calculus facts we have covered so far in class.Problem 3. Evaluate the following limits, or show they do not exist:a. limx→11 − x2+ 4x3sin(x)b. limx→0|x|sin(x)c. limx→0cos ex3− 1x − sin(x)!d. limx→3Zx3e−t2dt.e. limx→31x − 3Zx3e−t2dt.f. limx→0exp(x/ sin(x))Problem 4. Let us definef(x) = “the second digit after the decimal point in the decimal expansion for x”.Does limx→0f(x) exist? If so, what is it? If not, why not?Problem 5. A parallelogram has sides of length one meter that are hinged at the ends.At the moment when the height is√22m it is decreasing at 1 m/s. How fast is the anglechanging?Problem 6. Evaluate the following definite integralsa.Zπ0x cos(x) dxb.Z21x2ln x dx c.Z0−1x(1 + x)2012dxProblem 7. Evaluate the following indefinite integralsa.Z11 +√xdx b.Z√xe√xdxc.Zsin(θ)1 − cos2(θ)dθProblem 8. A particle at the end of a string spins in a circle about the origin. At theinstant that the particle is at the point (1/2 m,√3/2 m), the x coordinate is decreasing at arate of 1 m/s.a. Find the rate of change of the y coordinate.b. Find the rate of change of the angle θ.Problem 9. Some questions about continuity:a. If f (x) is continuous, is it necessarily true that (f(x))2is continuous?b. If f (x) is not continuous is it necessarily true that (f(x))2is not continuous?Problem 10. Consider the function f(x) = x2013+x+1. Show that there is a point c ∈ (0, 1)such that f0(c) = 2013.Problem 11. Find the first three non-zero terms in the Taylor series for f(x) = arctan(x)about the point a = 0.Problem 12. Compute the upper Riemann sum U4for the function f(x) = x3+ x2for theinterval [1, 2] with n = 4 subintervals.Problem 13. The largest box the United States Postal Service (USPS) will accept is onewhose combined length (longest dimension) and girth (circumference around the box per-pendicular to the length) is 108 inches. What is the largest possible box volume, in cubicfeet, that the USPS will accept?Problem 14. A metal drum in the shape of a right circular cylinder with no top must bebuilt to have surface area 3π ft2. What height h and base radius r will maximize the volumeof the cylinder
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