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

$x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - \frac{x^{7}}{7!} + \dots$ .

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

$\int_{0}^{t} x \cos (x) d x$ .

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

$E (x) = \int_{0}^{x} \frac{e^{t} - 1}{t} d t$

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

$F (x) = \int_{0}^{x} \ln (1 + t^{2}) d t$ .

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

$y^{'} (t) = y + b t$

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

$y^{'} (t) = y$ .