Test 1
1998, Fall

Version A

Mixture. 10 points.

We have 5 kg 30% concentrated solution. How much of 5% solution should be added to decrease its concentration to 15%?

Straight lines. 11 points.

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

Parabola. 12 points.

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

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

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

Transformations. 10 points.

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

Functions. 12 points.

Determine the domain of definition for the function f(x)=(Sqrt[(1/(-(x^2)+8 x-12))])+lnπ.

Inverse function. 12 points.

Find the inverse function of f(x)= (1/(((x+1))^3))-2!

Tangent line. 11 points.

Give the equation of the tangent line to the function f(x)= (1/(x+1))+2 at the point of x=1! Draw the graph of f(x) and the tangent line!

Derivative 1. 10 points.

Differentiate the following functions:

(1-2(Sqrt[x]))tanx,		((ln x)/((x^2)+1)).

Derivative 2. 12 points.

Differentiate the following functions:

(Sqrt[4 (x^2)-2 x+1])+(Sqrt[2]), 	((sine)^x)-cosπ.

Version B

Mixture. 10 points.

We have 60g 10% concentrated solution. How much concentrated solution should be added to obtain 100g 15% concentrated mixture?

Straight lines. 11 points.

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

Parabola. 12 points.

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

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

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

Transformations. 10 points.

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

Functions. 12 points.

Determine the domain of definition for the function f(x) = ln(6-x-2 (x^2)) + (Sqrt[e]).

Inverse function. 12 points.

Find the inverse function of f(x)=1- (8/(x^3))!

Tangent line. 11 points.

Give the equation of the tangent line to the function f(x)= (Sqrt[x+3])-1 at the point of x=1! Draw the graph of f(x) and the tangent line!

Derivative 1. 10 points.

Differentiate the following functions:

2 (e^x)-1,		((cot x)/((Sqrt[x])+1)).

Derivative 2. 12 points.

Differentiate the following functions:

cos(2x-(x^2)) + sinπ, 		(e^((Sqrt[x])-x)) - (e^2).