1. Find these limits.
(a) limt->0 (sin(4t)-t1)/(e7t-1).
(b) limx->0 x5x
(c) limn->\infty \pi/n*\sum_{k=1}^n sin(\pi*k/n)
2. Find these integrals
(a) \int cos(2-4x) dx
(b) \int x/(x2+1) dx
3. Find the area of the region bounded below by the curve y=x2 and above by the curve y=4-x2.
4. An cylinder (of radius r, height h) without a top is to be built with fixed volume V. Find the dimensions of the cylinder which has minimal surface area.
5. True of False (If true give a proof, if false give a counterexample. An answer alone is worth zero.)
(a) If Newton's method is used to find a solution to x3=x+1 with initial guess x1=3/2, then the next iterate x2= 31/23.
(b) The Fundamental Theorem of Calculus shows that \int-1 1 1/x dx=0.
(c) A function of x, whose derivative is ex2 is \int_2x2 et dt.
6. Give a diagram which shows the appropriate areas and proves
\int14 e-x3 dx is less than 3/e.
7. Sketch the graph of y=x*ln(x) identifying all local max/mins, asymptotes and points of inflection.
1. Find these limits.
(a) limx->1 ln(x)/(x-1)
(b) limx->0(1+tan(2x))1/x.
(c) limn->\infty 1/n*\sumi=1n e1+i/n
2. Find these integrals
(a) \int0\pi/12 sec2(2x) dx
(b) \int25 ex (1+ex)1/3 dx
(c) \int arcsin(x) \sqrt(1+(arcsin(x))2)/\sqrt(1-x2) dx
3. Find the area of the region bounded below by the curve y=sin(x), above by the curve y=cos(x), from x=0 to x=\pi/4.
4. A rectangle, whose sides are parallel to the x- and y-axes, is inscribed in the ellipse x2/22+y2/1=1. Find the rectangle which has the largest area.
5. True of False (If true give a proof, if false give a counterexample. An answer alone is worth zero.)
(a) If f ''(a)=0, then a must be an inflection point of f(x).
(b) The derivative with respect to x of \intab f(x) dx is f(b).
(c) Let D=d/dx, and I=integrate from 0 to x. Then D(I(f(x)))=f(x), when f is continuous on [0,1], x between 0 and 1.
6. What is \int_12 g(x)5 g '(x) dx?
7. Sketch the graph of y=x2+1/x, identifying all local max/mins, asymptotes and points of inflection.
1. Find (a) limx->0 (1+tan(2x))csc(3x)
(b) limn->\infty 1/n*\sumi=1n (i/n)C, where C is a positive constant.
2. Find (a) \int 2u/((3+u2)\sqrt(u2+2)) du
(b) \int12 sin(cos(sin(x)))*sin(sin(x))*cos(x) dx
3. Prove that the area inside the ellipse x2/a2+ y2/b2=1 is \pi*a*b.
4. A right circular cylinder is inscribed in a sphere of radius a. Find the cylinder which has the largest volume.
5. True or False (watch out, may be tricky).
(a) An antiderivative of sin(2x) is 1/2*(sin2(x)-cos2(x)).
(b) \intxx2 f '(t) dt= f(x)2-f(x), where f '(t) is continuous.
(c) Which is larger? A=\int 01 ex/(1+e2x) dx, or B= \int01 \sqrt(1-x2) dx?