Test 1
2009, Fall

Mixture. 5 points.

How much of 20% and 80% concentrated solutions should be mixed to obtain 2 kg solution of 35% concentration?

Algebraic and logarithmic rules. 10 points.

Simplify the following expression:

(( ((y^3)/(x^2)) +y (Sqrt[x]))/((y^(1/3))x));	((5  (Sqrt[5]))^(1/3));	((1/81)^(1/3));	  (0.01^(lg 111));		(Subscript[log, 3]) (1/81).

Straight lines. 4 points.

Write the slope, the equation and plot the graph of the straight line passing through the points  (1,-2) and (3,1).

Parabola. 6 points.

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

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

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

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

Inverse function and elementary transformations. 5 points.

Find the inverse function (OverBar[f])(x) to f(x)=(Subscript[log, 3])(x-2)+1.

Logarithmic coordinate system. 5 points.

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

Plot[Log[10,10^(-4x/7+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->{Thickness[0.01]}]

Logarithmic coordinate system. 5 points.

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

Derivative. 2+3+4+6 points.

Find the derivative of the following functions:

(x^(1/3))-(3^x)+π ;      (((Subscript[log, 2])x)/(x^2));  (x^(1/5)) (2^x);       (e^(-((((x-1))^2)/4))).

Derivative graphically. 10 points.

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

Abs'[x_]=Sign[x];
Plot[  1.2(x+2.5)(x+1)^3 x(x-1)^2 (3^(- 0.5x^2)),{x,-5,1.3},PlotRange->All,AspectRatio->Automatic,PlotLabel->"f(x)"]

Tangent line. 4+6 points.

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

Graphical study of the 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[  1.2(x+2.5)(x+1)^3 x(x-1)^2 (3^(- 0.5x^2)),{x,-5,1.3},PlotRange->All,AspectRatio->Automatic,PlotLabel->"f'(x)"]

Investigation of functions. 10 points.

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

Mixture. 5 points.

What (salt or water) and how much of it do we have to mix to 20g salt solution of  50% concentration in order to obtain 55% concentrated solution.

Algebraic and logarithmic rules. 10 points.

Simplify the following expression:

((  ((x^4)/(y^2))+y (x^(-2)))/((Sqrt[y])x));   	 ((2(2^(1/3)))^(1/( 4))); 	 ((-64)^(1/3));	  (100^(lg 3)); 	 (Subscript[log, 1/2]) 8;

Straight lines. 4 points.

Write the equation and plot the graph of the straight line with the slope m=-(1/4) and passing through the point (1,2).

Parabola. 6 points.

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

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

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

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

Inverse function and elementary transformations. 5 points.

Find the inverse function (OverBar[f])(x) to f(x)=(2^(x+1))-2.

Logarithmic coordinate system. 5 points.

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

Plot[Log[10,10^(3x/4-2)],{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]}]

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.

Derivative. 2+3+4+6 points.

Find the derivative of the following functions:

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

Derivative graphically. 10 points.

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

Abs'[x_]=Sign[x];
Plot[  2x (x-1)^3 (x+1)^2(x-2) (3^(- 0.7x^2)),{x,-1.2,4},PlotRange->All,AspectRatio->Automatic,PlotLabel->"f(x)"]

Tangent line. 4+6 points.

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

Graphical study of the 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[ 2x (x-1)^3 (x+1)^2(x-2) (3^(- 0.7x^2)),{x,-1.2,4},PlotRange->All,AspectRatio->Automatic,PlotLabel->"f'(x)"]

Investigation of functions. 10 points.

Consider the function f(x)= (x^3) (e^x). Find the extremal points and the intervals where the 	function is monotone increasing.