Test 1
2002, Fall

Version A

Mixture. 5 points.

We have 2 kg 30% concentrated solution. How much of 10% solution should be added to decrease its concentration to 25%?

Algebraic rules. 1+2+2 points.

Simplify the following expressions:

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

Evaluation. 4 points.

Evaluate the following expressions!

((1/27)^(1/3)), 	(Subscript[log, 4]) (1/64), 	(10^lg3), 		(Subscript[log, (1/3)])9.

Elementary functions. 6 points.

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

(e^(-x)), 	(1/(x+2)), 	(Subscript[log, 0.5])x, 	tanx, 	sinx, 	(x^(1/3)).

Straight lines 1. 5 points.

Plot the following straight lines and find their intersection point!

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

Straight lines 2. 5 points.

A straight line passing through the point (6, -1) intercepts the y-axis at y=2. Give the equation of the straight line and draw its graph!

Parabola. 10 points.

Consider the following parabola:

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

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 (Sqrt[x]) are required to get the graph of 2-(Sqrt[x+3])? Make a sketch of the steps!

Functions. 10 points.

Determine the domain of defination for the function f(x) = ln(-2+x+(x^2)) + (1/x)!

Inverse function. 10 points.

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

Tangent line. 10 points.

Give the equation of the tangent line to the function f(x) = (x^3) -(x^2)- 4x + 2 at the point x=-2! Draw the graph of f(x) and the tangent line.

Derivative. 10 points.

Differentiate the following functions!

f(x) = (1-(1/x)) sin x, 	g(x) = ((ln x)/((x^3)+x)).

Draw the derivative of the following function!