Test 1
1999, Fall

Version A

Mixture. 10 points.

We have 6 kg 20% concentrated solution. How much of 12% solution should be added to decrease its concentration to 15%?

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

Sketch (draw) the graph and give the domain of the following functions!

(x^2),	(x^3), 	(1/x), 		sinx, 	tanx, 	 (Subscript[log, 2])x, 		(3^(-x)).

Algebraic rules. 5 point.

Find the following values:

((sinπ)/3), 	((tanπ)/4), 	(Subscript[log, 2]) (1/16), 	(10^lg3).

Straight line. 8 points.

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

Functions. 5 points.

Find the common points of the functions:

2x-1, 	(x^2)-4x-1.

Parabola. 12 points.

Determine the peak point equation and (if there exist real roots) the factorized equation of the parabola:

y= (x^2)-3x-4.

Give the coordinates of the peak point and the intersepts with the axes, and draw the graph of the parabola!

Transformations. 8 points.

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

Logarithmic coordinate system. 10 points.

Draw the logarithmically scaled interval [0, 1]!

Plot the function f(x) = (10^(-3x+1)) and the points (1, 2), (2, 4) in a logarithmic coordinate system!

Inverse function. 10 points.

Find the inverse function of f(x) = (((x-2))^3) - 1! Plot the graphs of f(x) and (f^(-1))(x)!

Limits 1. 8 points.

Find the following limits:

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

Limits 2. 14 points.

Find the following limits:

(lim_(x->-3))((-6+x+(x^2))/(9+6 x+(x^2))), 	(lim_(x->3))((-3-2 x+(x^2))/(3-4 x+(x^2))).