Test 2
2008, Spring

Mixture. 5 points.

How much of 15% and 60% concentrated solutions should be mixed to obtain 1 kg solution of  50% concentration?

Exponential and logarithmic rules. 6 points.

Simplify the following expression:

(32^(1/5));		(Subscript[log, 3]) (Sqrt[(1/27)]); 	(e^(1+ln 3)).

Algebraic and logarithmic equations. 3+3+3 points.

Solve the following equations:

(Subscript[log, 3]) (3x-1) =2; 	 (3^(x-2))=(3^(1/5))	 (Subscript[log, 2])((Subscript[log, 3])(x-1)) =2.

Elementary functions. 5+5+5 points.

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

(Subscript[log, 0.5])x;     (2^(-x+1));        (Sqrt[x-2])+1.

Logarithmic coordinate system. 5 points.

Write the formula for the function given in the following figure. Explain your solution.

p2=LogPlot[(10^(3x/5-2)),{x,-1,5},GridLines->{Automatic,Automatic},PlotRange->{0.001,100},AspectRatio->4/6];
p3=LogLogPlot[(x^(-1/3)),{x,0.0001,10},GridLines->Automatic,AspectRatio->2/4,PlotRange->{{0.0001,10},{0.1,50}}];
Show[GraphicsArray[{p2,p3}]]

Logarithmic coordinate system. 5 points.

Draw the 10 based log scale on the interval [1,10] 
Plot the function y=(10^(2-0.5x)) in a logarithmic system.

Derivaitve. 4+5+6 points.

Find the derivative of the following functions:

3(x^2)-((x-1)^(1/3))+ln(x);      (Sqrt[x]) (2^x);      (1/(1+(e^x))).

Derivative graphically. 10 points.

Plot the derivative of the following function f(x).  Mark the local extremal places and points of inflection.

Plot[ (x+3) (((x-2))^2) 1.5^x,{x,-20,3.2},PlotRange->All]

Tangent line and investigation of functions. 5+15 points.

Give the equation of the tangent line to the function  f(x)= (x^3)+(x^2)-x-2 at the point (Subscript[x, 0])=2. 
Do total investigation on it: monotonicity, concavity, maxima, minima, inflection points. Draw table!

Indefinite integral, elmentary methods. 5+5 points.

Find the following integrals using elemi integration methods:

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