Test 1
2006, Fall

Mixture. 6 points.

We have 0.1 kg 40% concentrated solution. How much of 60% solution should be added to increase its concentration to 45% ?

Algebraic and logarithmic rules. 15 points.

Simplify/evaluate the following expressions:

(1/6)-(1/4), 	((1/16)^(1/4)), 	(Subscript[log, 2])0.25, 	(((y^(-2))(x^2))/((y^4)(x^(-5)))), 	(1/(a+b)) + (1/(a-b)).

Straight lines. 8 points.

A straight line intercepts the y-axis at y = -5, and the x-axis at x = 1. Give the equation of the straight line and draw its graph.

Parabola. 10 points.

Find the peak-point equation and the factorized equation of the following quadratic function and draw its graph (with peak-point & intercepts):

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

Elementary functions. 9 points.

Give the domains and sketch the graphs of the following functions:

(2^(-x)), 	(1/(x-3)),	lgx.

Transformations. 10 points.

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

Logarithmic coordinate system. 10 points

Make a (semi-)logarithmic plot of the following function (plot as a straight line using logarithmic scale):

y = (2^(-2x+3))

Inverse function. 10 points.

Find and plot the inverse function of: 	y = f(x) = 2 + (1/(x-1))

Derivative. 12  points.

Differentiate the following functions.

1-2(x^2)+3lgx-4tanx, 	(e^x)(1-cosx), 	((Sqrt[x])/((x^2)+1)), 	ln(2+(2^x)).

Tangent line. 10 points.

Find the equation of the tangent line to the function f (x) at x0 = 3, and draw the graph of both. Give the domain of definition for f (x). Use the equation of the tangent line to approximate the value f (3.1).

f(x) = 4(Sqrt[x+1])

Mixture. 6 points.

We have 0.2 kg 40 % concentrated solution. How much of 20 % solution should be added to decrease its concentration to 25 % ?

Algebraic and logarithmic rules. 15 points.

Simplify/evaluate the following expressions:

(1/6)-(1/4), 	((1/16)^(1/4)), 	(Subscript[log, 2])0.25, 	(((y^(-2))(x^2))/((y^4)(x^(-5)))), 	(1/(a+b)) + (1/(a-b)).

Straight line. 8 points.

A straight line intercepts the y-axis at y = 5, and the x-axis at x = -1. Give the equation of the straight line and draw its graph.

Parabola. 10 points.

Find the peak-point equation and the factorized equation of the following quadratic function and draw its graph (with peak-point & intercepts):

y = -(x^2)+2x+3

Elementary functions. 9 points.

Give the domains and sketch the graphs of the following functions:

(3^(-x)), 	(Sqrt[x-3]), 	(Subscript[log, 3])x.

Transformations. 10 points.

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

Logarithmic coordinate system. 10 points.

Make a (semi-)logarithmic plot of the following function (plot as a straight line using logarithmic scale):

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

Inverse function. 10 points.

Find and plot the inverse function of: 	y = f(x) = 1 - (1/(x+2)).

Derivative. 12 points.

Differentiate the following functions:

2 - 2(x^(3 ))+ 4 (Subscript[log, 2])x - cotx, 	(2^x)(1+sinx), 	((e^x)/((Sqrt[x])+1)), 	ln(1+(x^2))

Tangent line. 10 points.

Find the equation of the tangent line to the function f (x) at x0 = 3, and draw the graph of both. Give the domain of definition for f (x). Use the equation of the tangent line to approximate the value f (3.1).

f(x) = 2(Sqrt[x-2])

Mixture. 6 points.

We have 0.3 kg 20 % concentrated solution. How much of 40 % solution should be added to increase its concentration to 35% ?

Algebraic and logarithmic rules. 15 points.

Simplify/evaluate the following expressions:

(1/6)-(1/4), 	((1/16)^(1/4)), 	(Subscript[log, 2])0.25, 	(((y^(-2))(x^2))/((y^4)(x^(-5)))), 	(1/(x-a)) + (1/(x+a)).

Straight line. 8 points.

A straight line intercepts the y-axis at y = -1, and the x-axis at x = 5. Give the equation of the straight line and draw its graph.

Parabola. 10 points.

Find the peak-point equation and the factorized equation of the following quadratic function and draw its graph (with peak-point & intercepts):

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

Elementary functions. 9 points.

Give the domains and sketch the graphs of the following functions:

(3^(x-1)),	(Sqrt[x+1]), 	(Subscript[log, 2])x.

Transformations. 10 points.

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

Logarithmic coordinate system. 10 points

Make a (semi-)logarithmic plot of the following function (plot as a straight line using logarithmic scale):

y = (10^(-0.5x+1)).

Inverse function. 10 points.

Find and plot the inverse function of: 	y = f(x) = 2 - (1/(x+1)).

Derivative. 12 points.

Differentiate the following functions:

3-4(x^4)+2(Subscript[log, 3])x-5sinx, 	((e^x)+1)cotx, 	(x/((2^x)+1)), 	ln(3-(Sqrt[x])).

Tangent line. 10 points.

Find the equation of the tangent line to the function f (x) at x0 = 1, and draw the graph of both. Give the domain of definition for f (x). Use the equation of the tangent line to approximate the value f (1.1).

f(x) = 4(Sqrt[x+3])

Mixture. 6 points.

We have 0.4 kg 30 % concentrated solution. How much of 10 % solution should be added to decrease its concentration to 15% ?

Algebraic and logarithmic rules. 15 points.

Simplify/evaluate the following expressions:

(1/(6 ))- (1/5), 		((1/27)^(1/3)), 	lg0.01, 		(((a^2)(b^(-3)))/((b^4)(a^(-4)))), 	(1/(x-y)) - (1/(x+y)).

Straight line. 8 points.

A straight line intercepts the y-axis at y = 1, and the x-axis at x = -5. Give the equation of the straight line and draw its graph.

Parabola. 10 points.

Find the peak-point equation and the factorized equation of the following quadratic function and draw its graph (with peak-point & intercepts):

y = 2(x^2)+5x+2

Elementary functions. 9 points.

Give the domains and sketch the graphs of the following functions:

(2^(x+1)), 		(1/(x+3)), 		lnx.

Transformations. 10 points

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

Logarithmic coordinate system. 10 points.

Make a (semi-)logarithmic plot of the following function (plot as a straight line using logarithmic scale):

y = (10^(0.5x-1))

Inverse function. 10 points.

Find and plot the inverse function of: 	y = f(x) = 1 + (1/(x-2)).

Derivative. 12 points.

4+3(x^3)+2(Sqrt[x])-cosx, 		(1-(e^x))tanx, 		((10^x)/((e^x)+1)), 		ln((1/x) -1).

Tangent line. 10 points.

Find the equation of the tangent line to the function f (x) at x0 = 0, and draw the graph of both. Give the domain of definition for f (x). Use the equation of the tangent line to approximate the value f (0.1).

f(x) = 2(Sqrt[x+4])