1. (a) Find the derivative of the functionf(x) =1x+2 cos x+3 tan x+4 cot x+5 ln x+6ex+log7x+8x+9 arctan x+arcsin x(b) Let f (x) = x2. Find f0(2) by using only the definition of the derivative.(c) Geometrically, the definite integralRbaf(x) dx represents the area of a certainregion on the x − y plane that is related to the curve y = f(x).Using only this geometrical interpretationfinZ30(2x + 1)dx.2. Let ~a = h2, 1i and~b = h1, 3i(a) Find the angle between ~a and~b(b) Let P (= (1, 3), Q = (−1, 5) and S = (5, 7) be points in a plane. Find thefourth vertex of the parallelogram whose sides are~P Q and~P S.3. The position vector of a particle traveling on the x − y plane at time t is ~r(t) =ht, 8t − t2i, where t is measured in seconds and coordinates are in meters.(a) Find the particle’s average velocity vector during the time interval [0, 2].(b) Find the particle’s velocity vector, speed, and acceleration vector at time t = 1.(c) Find a non-parametric equation describing the curve that the particle passesby.4. Find the derivatives of the following functions.(a) f(x) = (x2+ x + 1)(x3− 3x2+ x + 1). [no simplification for answer](b) g(x) =x + 1x2+ 1.(c) p(x) = (1 + x4)10(d) q(x) = sin1xe2x5. (a) Evaluatelimh→0(x + h)2− x2h(b) Evaluate limx→01 − exsin x(c) Evaluate limx→1x − 1|x − 1|(d) Evaluate limx→01 + xx cos x−1x6. (a) Find the Riemann sum R4=Xi=14 f(ci)(xi− xi−1) forZ80x2dx with regularpartition points xi= 2i for i = 0, 1, 2, 3, 4, and the middle point rule: ci=12(xi−1+ xi).(b) Evaluate the definite integralZ80x2dx.(c) Evaluate the indefinite integralZ x2+2x+ 3 cos x +4√1 − x2+51 + x2!(d) Find the derivative of the function F (x) =Zx0t2et2dt.7. (a) Use a linear approximation or a differential for the function f(x) = x1/3ata = 1000 to find an approximation to3√1003 −3√1000.(b) Let Use the Newton’s Method to find a rational number that approximates thepositive root to x2− 2 = 0.8. The derivatives of the function f (x) = xe−x2/2are calculated as followsf0(x) = (1 − x2)e−x2/2, f00(x) = x(x2− 3)e−x2/2(a) Find the intervals where f is increasing or decreasing. Also find points of localor global minimum or maximum.(b) Find intervals where f is concave up or concave down. Also find points ofinflection.(c) Find any horizontal asymptotes.(d) Sketch the curve of y = f (x) for −∞ < x < ∞.9. A box with a square base, rectangular sides, and open top must have a volume of1000 cm3. The material for the base costs $4/cm2and that for the sides $2/cm2.Find the dimensions of a box that minimizes the cost of material used.10. A swimming pool of dimension 100(m) × 200(m) and horizontal bottom is drainedat a rate of 2 m3/min. Find the rate of decreasing of the depth of the water in thepool.11. (a) Let q(x) = xx. Using the logarithmic differentiation technique, find q0(x).(b) Let y = y(x) be implicitly defined by y3+ xy = 1. Using the implicit differen-tiation technique, find y0(x).(c) Find the equation of the line that has slope 3 and is tangent to the curve givenparametrically by x = t2+ 1, y =
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