Lecture 24

Faculty of Science; Department of Mathematics
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Readings For This Lecture

Keshet Course Notes:

Chapter 10, pages 208 to 220

Summary

Group 5: Add a summary of the lecture in this space. Include examples, discussion, and links to external sources, if desired.

Exercises

1. The Taylor series for the function $\sin(x)$ is given by

$x-{\tfrac {x^{3}}{3!}}+{\tfrac {x^{5}}{5!}}-{\tfrac {x^{7}}{7!}}+\cdots$ .

Find the Taylor series of $\cos(x)$ by differentiating this function.

(x^2)/2 -(x^4)/(4*3!)+(x^6)/(6*5!)...

2. Find the Taylor series about $x=0$ of

$f(x)=x\cos(x)$ .

Use this result to evaluate the integral

$\int _{0}^{t}x\cos(x)dx$ .

Verify your answer using integration by parts.

3. Find the Taylor series about $x=0$ of

${\frac {e^{x}+e^{-x}}{2}}$ .

4. Find the Taylor series about $x=0$ of

${\sqrt {x}}\sin({\sqrt {x}})$ .

5. Approximate the value of ${\sqrt {110}}$ using the Taylor expansion of ${\sqrt {x}}$ about $x=100$ .

6. Evaluate the integral

$E(x)=\int _{0}^{x}{\frac {e^{t}-1}{t}}dt$

using Taylor series. Find the Taylor series of $E(x)$ .

7. Consider the exponential function $y=e^{x}$ . Write down a Taylor Series series expansion for this function. Divide both sides by $x^{n}$ . Now examine the terms for the quantity $e^{x}/x^{n}$ . Use this expansion to argue that the exponential function grows faster than any power function.

8. Find a Taylor series expansion about $x=0$ of

$F(x)=\int _{0}^{x}\ln(1+t^{2})dt$ .

9. Use a Taylor series representation to find the function $y(t)$ that satisfies the differential equation

$y'(t)=y+bt$

with $y(0)=1$ . This type of equation is called a non-homogeneous differential equation. Show that when $b=0$ your answer agrees with the known exponential solution of the equation

$y'(t)=y$ .