Test 1
2008, Fall

Mixture. 5 points.

How much of 25% and 80% concentrated solutions should be mixed to obtain 1 kg solution of  35% concentration?

Algebraic and logarithmic rules. 5 points.

Simplify the following expression:

(( ((y^2)/(x^3)) +y (x^(-3)))/((y^(1/4))x));	(Sqrt[3  (3^(1/3))]);		(Subscript[log, 2]) (1/32);	  (100^(lg 111));		(Subscript[log, 1/8]) 4.

Elementary functions. 1+1+2+3+3 points.

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

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

Straight lines. 3 points.

Write the equation and plot the graph of the straight line with the slope m=(2/3) 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: 1+2 x-(x^2).

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. Mark and interval on the figure where the function is one-to-one. Plot the function -0.5f(x-3).

pf=Plot[{Abs[x+2](x)^2},{x,-3,1},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=Plot[Log[10,10^(3x/5-1)],{x,-2,4},AspectRatio->Automatic,
Ticks->{Automatic,Table[{Log[10,10.^i],10.^i},{i,-4,4}]},GridLines->Automatic];
p3=ParametricPlot[{Log[10,x],Log[10,10x^(-1/2)]},{x,0.005,500},AspectRatio->Automatic,
Ticks->{Table[{Log[10,10.^i],10.^i},{i,-4,4}],Table[{Log[10,10.^i],10.^i},{i,-4,4}]},GridLines->{Table[Log[10,10.^i],{i,-4,8}],Table[Log[10,10.^i],{i,-4,8}]},PlotRange->All];
Show[GraphicsArray[{p2,p3}]]

Derivative. 2+3+4+5+6 points.

Find the derivative of the following functions:

(x^2)-(Sqrt[x+1])+cos(x);  ((sin(x))/((Subscript[log, e]) (x)));   (e^(Sqrt[x]));   (x^(1/3)) (Subscript[log, 2])(x);    ((e^((Subscript[log, e])(x)+x))/(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+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)+3(x^2)-2 at the point (Subscript[x, 0])=-2 and at the inflection points.

Mixture. 5 points.

What (salt or water) and how much of it do we have to mix to 15g salt solution of  35% concentration. to obtain 50% concentrated solution.

Algebraic and logarithmic rules. 5 points.

Simplify the following expression:

((  ((y^4)/(x^2))+y (x^(-1)))/((Sqrt[x])y));   	 ((2(Sqrt[2]))^(1/( 5))); 	 (Subscript[log, 9]) (1/3);	  (100^(lg 3)); 	 (Subscript[log, 1/3]) 9;

Elementary functions. 1+1+2+3+3 points.

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

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

Straight lines. 3 points.

Write the equation and plot the graph of the straight line with the slope m=-(2/3) 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: 2+4 x-(x^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. Mark and interval on the figure where the function is one-to-one. Plot the function -0.5f(x-3).

pf=Plot[{Abs[x+2]((x)^2+0.5)},{x,-2.5,1},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=Plot[Log[10,10^(-3x/5+1)],{x,-1,5.5},AspectRatio->Automatic,
Ticks->{Automatic,Table[{Log[10,10.^i],10.^i},{i,-4,4}]},GridLines->Automatic];
p3=ParametricPlot[{Log[10,x],Log[10,10x^(1/2)]},{x,0.005,200},AspectRatio->Automatic,
Ticks->{Table[{Log[10,10.^i],10.^i},{i,-4,4}],Table[{Log[10,10.^i],10.^i},{i,-4,4}]},GridLines->{Table[Log[10,10.^i],{i,-4,8}],Table[Log[10,10.^i],{i,-4,8}]},PlotRange->All];
Show[GraphicsArray[{p2,p3}]]

Derivative. 2+3+4+5+6 points.

Find the derivative of the following functions:

(x^3)+(Sqrt[x+3])+cot(x); ((sin(x))/((Subscript[log, e]) (x)));   (Sqrt[(e^x)]);    (Sqrt[x]) (2^x);         (Subscript[log, e])((x^2) (e^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+3) (((x-1))^2) (E^(- 0.4 Abs[x+1]^1.3)),{x,-12,10},PlotRange->All]

Tangent line. 10 points.

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