Test 2
2011, Fall

Elementary antiderivatives. 4+6 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!

Integration by parts and by substitution. 15 points

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

Graphical antidifferentiation. 10 points

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

Column[{Plot[Evaluate[0.3x+(Cos[Pi x/2])],{x,-3,3},AspectRatio->Automatic,AxesLabel->{x,y}],
Plot[0,{x,-3,3},AspectRatio->Automatic,AxesLabel->{x,y},PlotStyle->White,PlotRange->{-2,2}]}]

Definite integral graphically, 5 points

f:=Interpolation[{{-4,2},{-2,2},{0,0},{1,-2},{3,1},{5,1}},InterpolationOrder->1]
Plot[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

Numeric Integral, 5 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_]:=-0.3(x+2)(x)^2(x-2)+1
Plot[2-f[-x],{x,-1.2,2.1},PlotStyle->{Thickness[0.01]},PlotRange->{-0.5,1.5},GridLines->{Range[-1,2,0.5],None},AspectRatio->Automatic]

Area, 10 points

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

f(x)=-(x^3)+x  and g(x)=-2x

Variation of a function, 20 points

The velocity of a moving body is described by the equation s'(t)=v(t)= (1-t) (e^(-t)) (mg/s) (s(t) denotes the position of the body at t )
a) Find the antiderivative of  v(t). (7 pts)
b) Calculate the average velocity over the interval [0,2]. (5 pts)
c) Determine the formula for s(t) under the initial condition s(0)=1. (3 pts)
d) Find the maximum of s(t) (5 pts)

Infusion, 15 points

The speed equation of a drug dosing by infusion reaction is the differential equation m'(t)=2-m (t), where m (t) denotes the mass of the drug in the body at the instant t.
a) Draw the direction field and find the equilibrium (5 pts).
b) Find the solution if the initial value is m(0)=0 (10 pts).

Elementary antiderivatives. 4+6 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!

Integration by parts and by substitution. 15 points

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

∫sin(x) (x- (1/(1+cos(((x))^2))) )dx

Graphical antidifferentiation. 10 points

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

Column[{Plot[Evaluate[x^2(Sin[Pi x])],{x,-1.1,1.1},AspectRatio->Automatic,AxesLabel->{x,y}],
Plot[Evaluate[x^2(Sin[Pi x])],{x,-1.1,1.1},AspectRatio->Automatic,AxesLabel->{x,y},PlotStyle->White,PlotRange->{-1,1}]}]

Definite integral graphically, 5 points

f:=Interpolation[{{-3,1},{-1,0},{0,0},{2,-2},{4,1}},InterpolationOrder->1]
Plot[f[x],{x,-3,4},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

Numeric Integral, 5 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_]:=-0.3(x+1.5)(x)^2(x-2)+1
Plot[f[x],{x,-1.2,2.1},PlotStyle->{Thickness[0.01]},GridLines->{Range[-1,2,0.5],None},AspectRatio->Automatic]

Area, 10 points

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

f(x)=(x^3)-x  and g(x)=2x

Variation of a function, 20 points

The rate of change of the mass m(t) of a material over time t  is given by the function m'(t)=v(t)= -(t/(4(((1+(t^2)))^3)))(g/s)
a) Find the antiderivative of  v(t). (10 pts)
b) Calculate the average speed over the interval [0,3]. (5 pts)
c) Determine the formula for m(t) under the initial condition m(0)=1. (5 pts)

Chemical reaction, 15 points

The speed equation of a chemical reaction the differential equation m'(t)=2(1-0.5m (t)) , where m (t) denotes the mass of the producted material.
a) Draw the direction field and find the equilibrium (5 pts).
b) Find the solution if the initial value is m(0)=0 (10 pts).