Test 1
2008, Fall

Exponential and logarithmic rules. 6 points.

Simplify the following expression:

(64^(1/3));		(Subscript[log, 3]) (1/27); 	(2^((Subscript[log, 2]) 8)).

Algebraic and logarithmic equations 3+3+3 points.

Solve the following equations:

(Subscript[log, 2]) (8x) =-2; 	 (3^(x-2))=(1/27);	 (Subscript[log, 2])((Subscript[log, 3])(x-1)) =2.

Elementary functions. 5+5+5 points.

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

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

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}]]

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.

Derivative. 4+5+6 points.

Find the derivative of the following functions:

(x^3)-(Sqrt[x-1])+ln(x);      (Sqrt[x]) sin(x);      (((1+(e^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-3) (((x+2))^2) 1.5^x,{x,-20,3.2},PlotRange->All]

Tangent line and investigation of functions. 5+15 points.

Give the equation of the tangent line to the function  f(x)= (x^3)+(x^2)-x-2 at the point (Subscript[x, 0])=0. 
Do total investigation on it: monotonicity, concavity, maxima, minima, inflection points. Draw table!