Test 2
2013, Fall

Version A

Theory.  5 points

The definition of Antiderivative and Indefinite integral. Relation between the antiderivatives.

Elementary antiderivatives. 6+8 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!

Antiderivative with given initial value.  6+4 points

Find the antiderivative of  f(x)=(((x^2)-2x )/(x^2)) for which F(2)=2. 
Find the extremal points of  F(x).

Integration by parts and by substitution. 8+8 points

Integrate the function using the method of substitution or by parts.

∫(e^x)(Sqrt[(e^x)+1]) dx ; 		∫ x (((ln(x)))^2)dx.

Graphical antidifferentiation. 10 points

The figure below shows the derivative of the function F(x).
a) Where does F(x) have maxima, minima and inflection points?
b) Plot the elements of the tangent field at the given grid-points. Draw the graphs of the possible functions F(x).

Clear[f,F]
f[x_]:=Evaluate[4x(x-1)];
Row[{Plot[0.5f[x],{x,-0.5,1.5},AspectRatio->Automatic,AxesLabel->{x,y},PlotLabel->TraditionalForm[F'[x]],
GridLines->{Range[-3,3,0.5],None},GridLinesStyle->Directive[Black]]
,"   "
Plot[0,{x,-0.5,1.5},AspectRatio->Automatic,AxesLabel->{x,y},PlotStyle->White,PlotRange->{-1.1,1.5},GridLines->{Range[-3,3,0.5],Range[-3,2,0.5]},GridLinesStyle->Directive[Black]]}]

Definite integral, 10 points

Calculate the following integral:

(∫_1^4)((2(x^3)-x)/x)dx

Definite integral graphically, 7 points

f=Line[{{-2,1},{-1,0},{0,0},{2,-2},{3,0}}];
Graphics[{Thickness[0.01],f},GridLines->{{-4,-3,-2,-1,0,1,2,3,4,5},{-4,-3,-2,-1,0,1,2,3,4,5}},GridLinesStyle->Directive[Black,Opacity[0.8]],AspectRatio->Automatic,Axes->True]

Numeric Integral, 7 pont

Sketch in the figure and calculate the left-side approximation of the integral of the function below on the interval [-2,1] using the marked subintervals.

Clear[f,g]
f[x_]:=-2 x+(x^2)+(x^3)
Plot[f[x],{x,-2,1},PlotStyle->{Thickness[0.01]},GridLines->{Range[-2,2,0.5],None},GridLinesStyle->Directive[Black,Opacity[0.8]],AspectRatio->Automatic,PlotRange->{-0.7,2.2}]

Area, 10 points

Calculate the absolute area enclosed by the graphs of the following functions:

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

Variation of a function, 16 points

The velocity of a moving body is described by the equation s'(t)=v(t)= (t^3)+sin(t) (m/s) . Here s(t) denotes the position of the body at t .
a) Find the indefinite integral of  v(t). (3 pts)
b) Calculate the average velocity over the interval [1,4]. (5 pts)
c) Determine the formula for s(t) under the initial condition s(0)=1. (4 pts)

Version B

Elementary antiderivatives. 6+8 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!

Antiderivative with given initial value.  10 points

Find the antiderivative of  f(x)=(x^2)(x-1) for which F(0)=1. 
Find the extremal places of  F(x).

Integration by parts and by substitution. 16 points

Integrate the function using the method of substitution or by parts.

∫sin(x) (x-(1/((cos^3)(x))))dx.

Graphical antidifferentiation. 10 points

The figure below shows the derivative of the function F(x).
a) Where does F(x) have maxima, minima and inflection points?
b) Plot the elements of the tangent field at the given grid-points. Draw the graphs of the possible functions F(x).

f[x_]:=6Evaluate[(-0.3(x-1)(x)^1(x+1.5))];
Row[{Plot[0.5f[-x],{x,-1.3,1.6},AspectRatio->Automatic,AxesLabel->{x,y}],"   ",
Plot[0,{x,-1.3,1.6},AspectRatio->Automatic,AxesLabel->{x,y},PlotStyle->White,PlotRange->{-1.1,2},GridLines->{Range[-3,3,0.5],Range[-2,2,1]},GridLinesStyle->Directive[Black]]}]

Definite integral graphically, 7 points

f:=Interpolation[{{-4,2},{-2,2},{0,0},{1,-2},{3,1},{5,1}},InterpolationOrder->1]
Plot[1-f[-x],{x,-4,5},PlotStyle->{Thickness[0.01]},GridLines->{{-4,-3,-2,-1,0,1,2,3,4,5},{-4,-3,-2,-1,0,1,2,3,4,5}},GridLinesStyle->Directive[Blue,Opacity[0.5]],AspectRatio->Automatic]

Definite integral, 10 points

Calculate the following integral.

(∫_2^4)(((Sqrt[x])+(x^2))/x)dx

Numeric Integral, 7 pont

Sketch in the figure and calculate the right-side approximation of the integral of the function below on the interval [-1,2] using the marked subintervals.

Clear[f,g]
f[x_]:=0.5-(-0.3(x+2)(x)^2(x-2))
Plot[f[x],{x,-1.2,2.1},PlotStyle->{Thickness[0.01]},PlotRange->{-1,1},GridLines->{Range[-1,2,0.5],None},AspectRatio->Automatic,GridLinesStyle->Directive[Black]]

Area, 10 points

Calculate the absolute area enclosed by the graphs of the following functions:

f(x)=(x^2) +x and g(x)=2(x^2)

Variation of a function, 16 points

The velocity of a moving body is described by the equation s'(t)=v(t)=(t^2)-t (m/s) (s(t) denotes the position of the body at t )
a) Find the antiderivative of  v(t). (3 pts)
b) Calculate the average velocity over the interval [1,4]. (5 pts)
c) Determine the formula for s(t) under the initial condition s(0)=1. (4 pts)
d) Find the extremal value of s(t) (3 pts)