Test 1
2013, Spring

Version A

Part. Elementary problems

Mixture. 5 points.

How much of 25% and 60% concentrated solutions should be mixed to obtain 15 kg solution of  45% concentration?

Algebraic and logarithmic rules. 1+1+1+2  points.

Simplify the following expression:

((1/128)^(1/5));	(((Sqrt[x]) y (z^2))/(x (y^(1/2))z));   (0.1^(lg 2013));	   (Subscript[log, x]) (1/(x^2)).

Straight lines. 4 points.

Write the equation of the straight line passing through the point  (-1,-2) with the slope m=2. Plot the graph, and calculate the zero point of it.

Parabola. 6 points.

Write the zero points and the peak point, and plot the graph of the following function: -1+x+2 (x^2). What is the factorized form of the equation?

Elementary functions. 2+2+2+4 points.

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

-2x+1;  (1/x);   (Subscript[log, 2])x;     0.5((x-1)^(1/3))+2.

Transformations. 5 points.

Plot -0.5f(x-1)-1 if  f(x) is given by the following figure:

paf=Plot[Min[2Abs[x-1],1]-1,{x,-2,2},
AspectRatio->Automatic,PlotRange->{-1,1}]

Half-life. 5 points.

Find the half-life  of f(x)=(5^(-2x+1)).

Logarithmic coordinate system. 5 points.

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

Logarithmic coordinate system. 5 points.

Write the formula for the functions given in the following figures. Explain your solution.

Plot[Log[10,10^(2x/3-2)],{x,-1,7},AspectRatio->Automatic,
Ticks->{Table[{i,i},{i,-4,10}],Table[{Log[10,10.^i],10.^i},{i,-5,4}]},GridLines->{Table[{i,Blue},{i,-4,10}],Table[{Log[10,10.^i],Blue},{i,-5,4}]},PlotStyle->{Thickness[0.01]},GridLinesStyle->Directive[Opacity[0.2]]]

Part. Derivative and applications

Theoretical question (5 points)

The relation between the monotonicity of f(x) and the properties of f'(x).

Rules of derivative. 2+3+4+6 points.

Find the derivative of the following functions:

(1/(x^3))+(Subscript[log, 3]) (x)-e ;       (x^(1/4)) sin(x);        ln(-2(x^3)+x);      ln(((e^x)/x)) .

Derivative graphically. 7 points.

Plot the derivative of the following function f(x). Draw the graph of the derivative.

Abs'[x_]=Sign[x];
Plot[  x(x-1) (x+1.5)^3  (1.5^( -x^2)),{x,-5.,1},PlotRange->All,AspectRatio->Automatic,PlotLabel->"f(x)"]

Tangent line. 6 points.

Give the equation of the tangent line to the function  f(x)= 2(x^3)+(2/x)+1 at the point (Subscript[x, 0])=-1.

Graphical study of derivative. 7 points.

The following curve is the graph of the derivative of f(x), i.e. f'(x). Mark the positions of maxima, minima, and inflection points of  f(x). Explain the answer!

Plot[  x(x-1) (x+1.5)^3  (1.5^( -x^2)),{x,-5.,1},PlotRange->All,AspectRatio->Automatic,PlotLabel->"f'(x)"]

Investigation of functions. 10 points.

Consider the function f(x)= 12 x-3 (x^2)-2 (x^3). Investigate the behavior of the function (monotonicity, convexity, extremal points, inflection points)