Test 2
2004, Fall

Version A

Limits. 10 points.

Find the following limits!

(lim_(x->∞)) ((2 (x^3)-4 x-1)/(3 (x^4)-3 (x^(-1)))),	 	(lim_(x->2))((x+1)/((x-2) (x-3))).

Tangent line. 5 points

Find the equation of the tangent line of the function f(x) = (x^5) - (x^2) +2x at (Subscript[x, 0])=1!

Graphical derivative. 10 points.

Draw the derivative of the following function!

Derivative. 3+6+6 points.

Differentiate the following functions!

(x^4) - 2(x^2) - π, 	((e^x)/((x^2)-x+1)), 		ln(sinx).

Investigation of a function. 20 points.

Do the total examination (domain, zeros, monotonicity, extremal points, concavity, inflection points, limits at plus and minus infinity, graph) of the following function!

f(x) = (x/(1+(x^2)))

Elementary antiderivatives. 5+5+10 points.

Find the following antiderivatives using elementary methods!

∫((x^4) - (x^(1/3)))xdx, 	∫cos3x + (Sqrt[2])dx, 	∫ (x/(x(x-2))) dx.

Integration by substitution. 10 points.

Use the method of integration by substitution!

∫ (1/(x(lnx))) dx

Variation of a function. 10 points.

Let the velocity of a moving body be v(t) = (1/(t+1)) and the initial position is s(0) = 0. Find the distance function s(t)!