Problems 1

1.

Check that

$$\frac{d}{dx} \int_1^x \cos(y)^{20} dy = cos(x)^{20}.$$

2.

Let A be the circle of radius 2 centered at (-1,0). Let B be the circle of radius 2 centered at (1,0). Find the points at which A and B intersect.

3.

Is it true that $\ln(1+x) \leq x$ for all $x \geq 0$? Is it true that $\ln(1+x) \leq x - \frac{x^2}{2} + \frac{x^3}{3}$ for all $x \geq 0$? Are you just guessing or can you prove it?

4.

What is the greatest common divisor of x³+1 and x²+3x+2?

5.

Find all integer solutions of the equation 5x+7y=37 (both x and y should be integers).

6.

Solve the differential equation $y''(x) + y(x) = \sin(x)$ with initial conditions y(0)=1, y'(0)=0. Plot your solution.

7.

Use the commands seq and ithprime to generate a list of the first 20 primes. Compute the sum of these primes, and give its integer factors.

8.

Draw a graph showing both $\cos(x)$ and its fifth Taylor polynomial (that is, $1-\frac{1}{2!}x^2+\frac{1}{4!}x^4$) for x between -4 and 4. How many terms do you seem to need to get good agreement in this range. Hint: use a variation of the command convert(taylor(cos(x),x,5),polynom) to make this work. Think of a suitable way to demonstrate that the approximation you have taken is good.