Test 1
2013, Fall

Version A

Part. Elementary problems

Mixture. 5 points.

How much of 30% and 50% concentrated solutions should be mixed to obtain 20 kg solution of  55% concentration?

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

Simplify the following expression:

(Sqrt[3  (27^(1/3))]);	(((Sqrt[x]) y (z^3))/(x (Sqrt[y])(z^2)));   (10^(-3lg 2));	(Subscript[log, x]) (1/x).

Straight lines. 6 points.

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

Parabola. 7 points.

Write the zero points and the peak point, and plot the graph of the following function: 3-4 x+ (x^2).

Elementary functions. 2+2+3 points.

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

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

Transformations. 5 points.

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

paf=Plot[-Min[Abs[-x+1],1],{x,-2,2},
AspectRatio->Automatic,PlotRange->{-1.1,1},PlotStyle->{Thick}]

Half-life. 5 points.

Read the half-life of the function given by the following graph:

Plot[3 2.5^(-x),{x,0,5},AspectRatio->Automatic, GridLines->Automatic,PlotRange->{0,3},PlotStyle->{Thick}]

Logarithmic coordinate system. 5 points.

Draw the 10 based log scale on the interval [1,10]. 
Plot the function 0.1*(10^(-2x+1)) 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[0.8]]]

Part. Derivative and applications

Rules of derivative. 5+7+8 points.

Find the derivative of the following functions:

(3/(x^2))-cos(x)+π ;      (x^(1/5)) (Subscript[log, 3])x;       (e^(ln(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[(2x(x-0.5) (x+1.5)^3  (1.5^(- x^2))),{x,-2.3,0.6},PlotRange->All,AspectRatio->Automatic,PlotLabel->"f(x)",PlotStyle->{Thick}]

Tangent line. 6 points.

Give the equation of the tangent line to the function  f(x)= (Sqrt[x])-x+1 at the point (Subscript[x, 0])=4.

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!

Abs'[x_]=Sign[x];
Plot[(2x(x-0.5) (x+1.5)^3  (1.5^(- x^2))),{x,-2.3,0.6},PlotRange->All,AspectRatio->Automatic,PlotLabel->"f'(x)",PlotStyle->{Thick}]

Investigation of functions. 10 points.

Consider the function f(x)=-3 (x^4)+2(x^2)-1. Investigate the behavior of the function (monotonicity, convexity, extremal points, inflection points)

Version  B

Part. Elementary problems

Mixture. 5 points.

What (salt or water) and how much of it do we have to mix to salt solution of 10% concentration in order to obtain 30 kg of 25% concentrated solution.

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

Simplify the following expression:

((4  (Sqrt[4]))^(1/3));  	(((Sqrt[y]) x (z^2))/(y (Sqrt[x])z));     (10^(-2lg 3));     (Subscript[log, x]) (Sqrt[x]).

Straight lines. 6 points.

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

Parabola. 7 points.

Write the zero points and the peak point, and plot the graph of the following function: -3-2x+ (x^2).

Elementary functions. 2+2+3 points.

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

(Sqrt[x]);   (3^(-x)); sin(x).

Transformations. 5 points.

Plot -2f(x)+1 if  f(x) is given by the following figure:

paf=Plot[Which[-1<x<0,Sin[Pi x],True, 0],{x,-2,2},
AspectRatio->Automatic,PlotRange->{-1.1,2},PlotStyle->{Thick}]

Half-life. 5 points.

Find the half-life  of f(x)=(4^(-x+2)).

Logarithmic coordinate system. 5 points.

Draw the 10 based log scale on the interval [1,10] 
Plot the function y=10*(10^(0.6x-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/5-2)],{x,-1,10},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[0.8]]]

Part. Derivative and applications

Derivative. 5+7+8 points.

Find the derivative of the following functions:

-(3/(x^4))-ln(x)+(Sqrt[2]) ;       ((sin(x))/(x^2));    (e^(-x+(Sqrt[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)^2  (1.5^(- x^2)),{x,-2,1.1},PlotRange->All,AspectRatio->Automatic,PlotLabel->"f(x)",PlotStyle->{Thick}]

Tangent line. 6 points.

Give the equation of the tangent line to the function  f(x)= -3(x^5)-2x+2 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!

Abs'[x_]=Sign[x];
Plot[-x(x-1) (x+1.5)^2  (1.5^(- x^2)),{x,-2,1.1},PlotRange->All,AspectRatio->Automatic,PlotLabel->"f'(x)",PlotStyle->{Thick}]

Investigation of functions. 10 points.

Consider the function f(x)=(x^4)+(x^3)-2. Investigate the behavior of the function (monotonicity, convexity, extremal points, inflection points).