Test 2
2003, Fall

Version A

Mixture. 5 points.

How much of 40% and 60% concentrated solutions should be mixed to obtain 10 kg solution of 45% concentration?

Elementary functions. 15 points.

Write the domain and plot the graph of the following functions!

(3^x)-1, 	(1/(1-x))+2, 		1+ (Subscript[log, 3])x, 	tan3x, 		((x+2)^(1/3)).

Inverse function. 10 points.

Find and plot the inverse function of f(x) = 1-(Subscript[log, 2]) x !

Derivative. 4+8+8 points.

Differentiate the following functions!

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

Tangent line. 10 points.

Find the equation of the tangent line of f(x) = (Sqrt[-2x+6(x^2)]) at (Subscript[x, 0])=1!

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!

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