Test 1
2011, Fall

Mixture. 5 points.

How much of 15% and 60% concentrated solutions should be mixed to obtain 20 kg solution of  30% concentration?

Algebraic and logarithmic rules. 5 points.

Simplify the following expression:

((y-((Sqrt[y])/x))/((x^2) (Sqrt[y])));	((-2  (Sqrt[8]))^(1/3));	((1/64)^(1/3));	  (100^(-lg 2));	(Subscript[log, x]) (1/x).

Straight lines. 4 points.

Write the equation of the straight line passing through the point  (1,-2) with the slope a=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: -(1/3)+((2 x)/3)+(x^2).

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

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

-2x+3;     (1/(x^2));      (x^(1/3));     (2^(-x));      2(((x-2))^3)-1.

Transformations. 5 points.

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

pf=Plot[{Abs[x-2](x)},{x,-1.5,2.5},PlotRange->{{-2,4},{-1.5,2}},
AspectRatio->Automatic]

Inverse function and elementary transformations. 5 points.

Find the inverse function (OverBar[f])(x) to f(x)=1+log(x-2).  Plot the graph of the inverse!

Logarithmic coordinate system. 5 points.

Draw the 10 based log scale on the interval [1,10]. 
Plot the function (10^(-2x+3/2)) 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+1)],{x,-2,8},AspectRatio->Automatic,
Ticks->{Table[{i,i},{i,-4,10}],Table[{Log[10,10.^i],10.^i},{i,-5,4}]},GridLines->{Table[{i,Black},{i,-4,10}],Table[{Log[10,10.^i],Black},{i,-5,4}]},PlotStyle->{Blue,Thickness[0.01]}]

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

Find the derivative of the following functions:

(1/(sin(x)))-π ln(x)+3 x ;      ((2^x)/(x^(1/3)));      ln(cos(x));     (e^((x^2)ln(x))) .

Derivative graphically. 10 points.

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

Abs'[x_]=Sign[x];
Plot[  4x(x-1)^2 (x+1)^3  (3^(- x^2)),{x,-2.5,1.5},PlotRange->All,AspectRatio->Automatic,PlotLabel->"f(x)"]

Tangent line. 5 points.

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

Graphical study of derivative. 5 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(x-1)^2 (x+1)^3  (3^(- x^2)),{x,-2.5,1.5},PlotRange->All,AspectRatio->Automatic,PlotLabel->"f(x)"]

Investigation of functions. 15 points.

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

Mixture. 5 points.

What (salt or water) and how much of it do we have to mix to salt solution of 5% concentration in order to obtain 12 kg of 20% concentrated solution.

Algebraic and logarithmic rules. 5 points.

Simplify the following expression:

(( (x-((Sqrt[z])/z)))/((Sqrt[x]) z));      ((-8  (Sqrt[2]))^(1/5));     (Subscript[log, 1/3]) 81;	      (0.01^(lg 3));    (Subscript[log, x]) (Sqrt[x]).

Straight lines. 4 points.

Write the equation and plot the graph of the straight line with the slope m=(1/2) and passing through the point (-1,-3).

Parabola. 6 points.

Write the zero points and the peak point, and plot the graph of the following function: (1/3)-((4 x)/3)+(x^2).

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

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

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

Transformations. 5 points.

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

pf=Plot[{Abs[x-2](-x)},{x,-0.5,2.5},PlotRange->{{-2,4},{-2,2}},
AspectRatio->Automatic]

Inverse function and elementary transformations. 5 points.

Find the inverse function (OverBar[f])(x) to f(x)=(e^(x-2))-3.

Logarithmic coordinate system. 5 points.

Draw the 10 based log scale on the interval [1,10] 
Plot the function y=(10^(3-2x)) 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^(-3x/7+2)],{x,-2,8},AspectRatio->Automatic,
Ticks->{Table[{i,i},{i,-4,10}],Table[{Log[10,10.^i],10.^i},{i,-5,4}]},GridLines->{Table[{i,Black},{i,-4,10}],Table[{Log[10,10.^i],Black},{i,-5,4}]},PlotStyle->{Thickness[0.01]}]

Derivative. 2+3+4+6 points.

Find the derivative of the following functions:

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

Derivative graphically. 10 points

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

Abs'[x_]=Sign[x];
Plot[ Evaluate[-(( 4x(x-1)^2 (x+1)^3  (3^(- x^2))))/.x->(-x)],{x,-1.5,2.5},PlotRange->All,AspectRatio->Automatic,PlotLabel->"f(x)"]

Tangent line. 5 points.

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

Graphical study of derivative. 5 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[ Evaluate[-(( 4x(x-1)^2 (x+1)^3  (3^(- x^2))))/.x->(-x)],{x,-1.5,2.5},PlotRange->All,AspectRatio->Automatic,PlotLabel->"f(x)"]

Investigation of functions. 15 points.

Consider the function f(x)=  ((x^2)+1) (e^x). Find the extremal points and the intervals where the function is monotone increasing.