Test 1
2014, Fall

Version A

Part. Elementary problems

Mixture. 5 points.

How much of 25% and 50% concentrated solutions should be mixed to obtain 150 mg solution of  40% concentration?

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

Simplify the following expression:

(Sqrt[5  (Sqrt[25])]);	(((Sqrt[z y]) +y (x^2))/(z (Sqrt[y])(x^2)));     (Subscript[log, 3])((3^(3x-1)));	(Subscript[log, (x^2)]) (1/(x^2)).

Straight lines. 6 points.

Write the equation of the straight line passing through the points  (-1,1) 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: 8-6 x+ (x^2).

Elementary functions. 2+2+3 points.

Write the domain and plot the graph of the following functions:   (x^3); (1/(x^2));  (2^(x-1))+1

Transformations. 5 points.

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

paf=Plot[Min[Abs[-2x+1],1],{x,-2,4},
AspectRatio->Automatic,PlotRange->{-1.1,1},PlotStyle->{Thick}]

Half-life. 5 points.

Find the half-life  of f(x)=0.1·(3^(-2x)).

Logarithmic coordinate system. 5 points.

Draw the 10 based log scale on the interval [1,10]. 
Plot the function 0.1·(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+1)],{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:

(2/(x^3))-(2^x)+(Sqrt[2]) ;      (((x^(1/3))+ (Subscript[log, 3])x)/x);       (((e^x)-(e^(-x)))/((e^x)+(e^(-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[-(4x^3 (x-0.8) (x+1.)(4^(- x^2))),{x,-2.,2},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)=(1/(x^2))-(x^2)+1 at the point (Subscript[x, 0])=2.

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[-(4x^3 (x-0.8) (x+1.)(4^(- x^2))),{x,-2.,2},PlotRange->All,AspectRatio->Automatic,PlotLabel->"f(x)",PlotStyle->{Thick}]

Investigation of functions. 10 points.

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