Lecture 25

Faculty of Science; Department of Mathematics
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Readings For This Lecture

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Summary

Group 6: This is a review class. Feel free to create examples and discussions on any topic in the course.

1. The Taylor Series for the function (1/(1+x^2)) around x=0 is 1-(x^2)+(x^4)-(x^6)+(x^8)

Use this fact to find the Taylor Series for arctan(x) around x=0

simply antidifferentiate the above series to make the taylor series for arctan(x)=

x-(x^3/3)+(x^5/5)-(x^7/7)...

2. Write down the first five terms in a Taylor Series for f(x)=e^x

e^x=1+x+(x^2/2)+(x^3/3)+(x^4/4)

3. Find the integral: integral(tan(x)sec^2(x)*dx

Let u=tan(x) du=sec^2(x)*dx (u^2)/2 +C sub back orginal function

tan^2(x)/2 +C

4. Evaluate the integral int(x*ln(x))*dx

let u=lnx du=1/x dv=x and v=(1/2)x^2

=ln(x)*(1/2)x^2-integral(1/2)x^2*(1/x)*dx

=(1/2)x^2lnx-(1/2)(x^2)/2

=(x^2)/2(ln(x)-(1/2)

5. Evaluate the definite integral integral from 0 to 1 of e^x/(e^2x +1)

let u=e^x du =e^x*dx

integral from 0 to 1 of du/(u^2 +1)

=arctan u evaluated at 1 and e

=arctan(e) -arctan (1)

=arctan(e)-pi/4

6. Find f(x) if x^2=1+int of 1 to x f(t)dt

d/dx(x^2)=d/dx(1+int of 1 to x f(t)dt)

2x=0+f(x)

therefore f(x)=2x

7. Determine if the integral from e to infinity of dx/(x(lnx)^3 converges or diverges. If it converges what is the value?

lim as R-infinity of integral of e to R dx/(x(lnx)^3 let u=lnx

lim as R--infinity [-1/2(lnx)^2]evaluated at e and R converges to (1/2)

8. Determine the value of the Reiman Sum of sigma k=10 N=50 of (k-2)

=(1+2+3+4...48)-(1+2+3..7)

=(48)(49)/2-(7)(8)/2

9. Find the Volume of y=(x)^(1/2) revolved around the y-axis.

V=pi*integral from 0 to 1 of (y^2)^2*dy

=pi*y^5/5 evaluated at 0 and 1 =pi/5

10. Determine an integral for the arc length of x-(2/3)(y-1)^(3/2) for the interval [2,5]

L=interal from 2 to 5 of the sqrt(1+f`(x)^2)dy L= int from 2 to 5 sqrt(y)*dy

11. The average value of f(x)=x^2 on the interval [b,1] is 1/3. What is b?

f=(1/(1-b)) int from b to 1 of x^2 dx

=(1/(1-b))*[1/3*x^3] evaluated at b and 1

=(1/(1-b))[(1/3)*(1^3-b^3)] foil this out and you get

(1/3)*(1+b+b^2)=1/3

b=0 or b=-1

12. Consider the equation f(x)=3+ integral of 0 to x (f(t)-)(f(t)-2)dt What is f(0)?

3+ int from 0 to 0

=3

13. Find the differential equation satisfied by f(x)

f`(x)=3+int from 0 to x(f(t)-1)(f(t)-2)dt]

=0+f(x)-1)(f(x)-2)

dy/dx=(y-1)(y-2) y(0)=3

14. Find all steady states of the differential equation dy/dt=y^3-y that are stable.

dy/dt=0 therefore y^3-y=0 y=0,1,-1.

By drawing a vector field, we see that y=0 is the only stable steady state.

What is the half life of a given radioactive substance pdf =P(t)=Ce^-kt

Find the MEDIAN! therefore

F(t)= integral from 0 to t of Ke^-ks ds

=-e^-ks evaluated at 0 and t = 1-e^-kt=(1/2)

(1/2)=e^-kt

2=e^kt

ln(2)=kt t=ln(2)/k