TEST 2 (MAT 370, REVIEW)

ASU ID:..................................NAME:..................................

1. Determing if the function is increasing or decreasing.

$$f(x)=x^3+5x+1$$

$$f(x)=-x^3-3x+1$$

$$f(x)= e^x$$

2. Prove that $x^3 + 5x + 1 =0$ has exactly one solution.

3. Prove that $x^3 + ax + b =0$ has exactly one solution for $a>0.$

4. Use the mean value theorem to prove that

$$\vert \cos(x) - \cos(y)\vert \le \vert x - y \vert$$

5. Find the limits.

$$\lim_{x\to 1} \frac{x^3-1}{x^2 - 1},\;\;\lim_{x\to \infty} (1 + \frac{3}{x})^x,\;\;\lim_{x\to 0 } x^x$$