Test 2
2006, Fall

Mixture. 6 points.

We have 5%, 20% and 40% concentrated solutions in stock. Which two types of available solutions should we mix (why?) in what quantities to obtain 600 g 10% concentrated mixture? (Calculate one solution only!)

Derivatives. 10 points.

Differentiate the following functions:

(sin2x/((x^3)+1)); 		sin((e^x)+1).

Investigation of a function. 16 points.

Do the total examination of the function (domain, intercepts, increasing / decreasing parts, minima / maxima, convex / concave parts, inflexions, graph of the function):

f(x) = -(x^3)+12x.

Elementary antiderivatives. 10 points.

Find the indefinite integral of the given function using fundamental integration. When necessary, first modify / simplify the function using identities to transform it into a linear combination of fundamental integrals!

∫((-ex+2(x^2)-3(x^3)+4(x^4))/(x^3))dx; 	∫(e^(-x))((e^(2x))-(e^x))dx.

Integration by substitution. 14 points.

Integrate the function using the method of substitution.

∫3(x^2)(e^(2(x^3)))dx; 	∫((e^tanx)/((cos^2)x))dx.

Integration by parts. 12 points.

Integrate the function using integration by parts.

∫3(x^2)lnxdx

Definite integral. 10 points.

Calculate the value of the definite integral.

(∫_0^(π/2))((1/π) + cos2x-sinx)dx

Area. 12 points.

Calculate the area enclosed by the two curves and also draw the graph!

f(x) = 1-(x^2) 	and 	g(x) = -x.

10 points.

Calculate the volume of revolution obtained by revolving around the x-axis the area between the curve of f (x) and the x-axis over the interval [a, b]. Draw a graph of the rotated area!

[a, b] = [-2, 0] 	and f(x) = 2(((x+1))^2)

Mixture. 6 points.

We have 5%, 20% and 50% concentrated solutions in stock. Which two types of available solutions should we mix (why?) in what quantities to obtain 400 g 30% concentrated mixture? (Calculate one solution only!)

Derivatives. 10 points.

Differentiate the following functions:

(tanx/((e^(2x))+1)); 		(e^(cosx-1)).

Investigation of a function. 16 points.

Do the total examination of the function (domain, intercepts, increasing / decreasing parts, minima / maxima, convex / concave parts, inflexions, graph of the function):

f(x) = (1/3)(x^3)-x.

Elementary antiderivatives. 10 points.

Find the indefinite integral of the given function using fundamental integration. When necessary, first modify / simplify the function using identities to transform it into a linear combination of fundamental integrals!

∫((((πx)^2)+3(x^3)-4(x^4)+5(x^5))/(x^4))dx; 	∫((((e^x)+2))^2)dx.

Integration by substitution. 14 points.

Integrate the function using the method of substitution.

∫4(x^3)(cos3x^4)dx; 	∫(((Sqrt[cotx]))/((sin^2)x))dx.

Integration by parts. 12 points.

Integrate the function using integration by parts.

∫2x cosxdx

Definite integral. 10 points.

Calculate the value of the definite integral.

(∫_(-π/2)^0)((2/π) + cosx-sin2x)dx

Area. 12 points.

Calculate the area enclosed by the two curves and also draw the graph!

f(x) = (x^2) 	and 	g(x) =2 -x.

10 points.

Calculate the volume of revolution obtained by revolving around the x-axis the area between the curve of f (x) and the x-axis over the interval [a, b]. Draw a graph of the rotated area!

[a, b] = [-2, 2] 	and f(x) = 4-(x^2)