Test 1
2003, Fall

Mixture. 5 points.

We have 3 kg 40% concentrated solution. How much of 60% solution should be added to increase its concentration to 45%?

Algebraic rules. 1+2+2 points.

Simplify the following expressions:

(1/3)-(1/4), 	((x^4)/(((y^2) (y^3))/(x^3))), 	(1/(a+b))+(1/((a^2)-(b^2))).

Evaluation. 4 points.

Evaluate the following expressions!

((1/81)^(1/4)), 	(Subscript[log, 2]) (1/32), 	lg(10^3), 		(Subscript[log, (1/3)])27.

Elementary functions. 6 points.

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

(3^(-x)), 	(1/(x-3))+1,	 	(Subscript[log, (1/3)])x, 		(x^(1/4)).

Straight lines 1. 5 points.

Plot the following straight lines and find their intersection point!

y+1 = 2x-1, 	-3y+1 = x-2.

Straight lines 2. 5 points.

A straight line passing through the point (1, 3) intercepts the x-axis at x=2. Give the equation of the straight line and draw its graph!

Parabola. 10 points.

Consider the following parabola:

y = (x^2)-4x+2.

Give the coordinates of the peak point and the intercepts with the axes and draw the graph!

Tranformations. 10 points.

Which transformation steps on the graph of the function (2^x) are required to get the graph of 3(1-(2^(-x)))? Make a sketch of the steps!

Inverse function. 10 points.

Find the plot the inverse function of f(x) = (e^(-1+2x))!

Logarithmic coordinate system. 10 points.

Draw a logarithmically scaled real line! Plot the function f(x) = (10^(2x-1)) and the points (1, 2), (2, 3), (3, 20) in a logarithmic coordinate system!

Limits. 10 points.

Find the following limits!

(lim_(x->∞)) ((2 (x^4)-5 x-1)/((x^4)-3 (x^2)-1)), 		(lim_(x->1)) ((2 x+1)/(1-5 x+2 (x^2))).

Tangent line. 10 points.

Give the equation of the tangent line to the function f(x) = (x^5) -(x^2)- 2x at the point x=1! Draw the graph of f(x) and the tangent line.

Derivative. 10 points.

Draw the derivative of the following function!