Test 1
2007, Fall

Mixture. 5 points.

How much of 20% and 70% concentrated solutions should be mixed to obtain 1 kg solution of  55% concentration?

Albebraic and logarithmic rules. 5 points.

Simplify the following expression:

(((y^(-2))+y (x^(-3)))/((y^(1/4))x));	(64^(1/4));		(Subscript[log, 4]) (1/64);	  (100^(lg 1001));		(Subscript[log, 1/8]) 0.5

Elementary functions. 2+2+3+4+4 points.

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

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

Inverse function. 10 points.

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

Plot. 5 points.

Let f(x) be given in the following figure. Plot the function 0.5f(x-2).

pf=Plot[{Abs[x-2](x)^2},{x,-0.8,2.5},PlotRange->{0,2.5},
AspectRatio->Automatic]

Logarithmic coordinate system. 5 points.

Draw the 10 based log scale on the interval [1,10] 
Plot the function y=(10^(1-3x)) in a logarithmic system.

Logarithmic coordinate system. 10 points.

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

p2=LogPlot[(10^(-2x/5+2)),{x,-1,5},GridLines->{Automatic,Automatic},PlotRange->{0.1,1000},AspectRatio->4/6];
p3=LogLogPlot[(x^(1/3)),{x,0.001,10},GridLines->Automatic,AspectRatio->3/4,PlotRange->{{0.0009,10},{0.01,10}}];
Show[GraphicsArray[{p2,p3}]]

Derivative. 5+5+5+5+5 points.

Find the derivative of the following functions:

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

Derivative graphically. 10 points.

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

Plot[- (x-2) (((x+2))^2) (E^(-0.3Abs[x]^1.2)),{x,-15,15},PlotRange->All]

Tangent line. 10 points.

Give the equation of the tangent line to the function  f(x)= (x^3)+6(x^2)-2 at the point (Subscript[x, 0])=0 and at the inflection points.

Mixture. 5 points.

What (salt or water) and how much of it do we have to mix to 10g salt solution of  25% concentration. to obtain 35% concentrated solution.

Albebraic and logarithmic rules. 5 points.

Simplify the following expression:

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

Elementary functions. 2+2+3+4+4 points.

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

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

Inverse function. 10 points.

Find the inverse function (OverBar[f])(x) to f(x)=ln(x-1)+2.  Plot both  f and  (OverBar[f]).

Plot. 5 points.

Let f(x) be given in the following figure. Plot the function f(2x)

pf=Plot[{Abs[x-2]((x)^2+0.5)},{x,-1,2.8},PlotRange->{0,2},
AspectRatio->Automatic]

Logarithmic coordinate system. 5 points.

Draw the 10 based log scale on the interval [1,10] 
Plot the function (10^(2x-3)) in a logarithmic system.

Logarithmic coordinate system. 10 points.

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

p2=LogPlot[(10^(2x/5)),{x,-1,5},GridLines->{Automatic,{0.001,0.01,0.1,1,10,100}},PlotRange->{0.1,1000},AspectRatio->4/6];
p3=LogLogPlot[(x^(-1/2)),{x,0.01,10},GridLines->Automatic,AspectRatio->2/3,PlotRange->{{0.009,10},{0.1,10}}];
Show[GraphicsArray[{p2,p3}]]

Derivative. 5+5+5+5+5 points.

Find the derivative of the following functions:

(x^5)+(Sqrt[x-3])+tan(x);     (Sqrt[x]) (e^x);         ln(x (e^x));         (1/((e^x)+1));        (Sqrt[-(2^x)])

Derivative graphically. 10 points.

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

Plot[ (x-1) (((x+1))^2) (E^(- 0.8 Abs[x]^1.2)),{x,-8,8},PlotRange->All]

Tangent line. 10 points.

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