Test 2
2005, Fall

Version A

Elementary functions. 10 points.

Plot the graph of the following functions!

(Sqrt[x]), 		(x^(1/3)), 		2(1- (e^(-x))), 	1- (1/(x-2)).

Tangent line. 5 points.

Find the equation of the tangent line of f(x) = (x^4) -2(x^2) +x-3 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!

3(x^5) - 4(x^3) - π + (1/x), 	(lnx/((x^3)-(x^2)+x)), 		(e^(1-(x^2))).

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) = (-4+x) (x^3)

Elementary antiderivatives. 5+5+10 points.

Find the following antiderivatives using elementary methods!

∫((2(x^3)-(x^(1/4))+x)/(x^2)) dx, 	∫cos5x + (2^(1/3))dx, 	∫ (x/((x+1)(x-3))) dx.

Integration by substitution. 10 points.

Use the method of integration by substitution!

∫ ((e^x)/((e^x)+1)) dx

Variation of a function. 10 points.

Let the velocity of a moving body be v(t) = (t^2) and the initial position is s(1) = 2. Find the distance function s(t)!