# Part 0: Calculus 1 Review

(Need a refresher on something? Try Paul’s Online Math Notes.)

### Review Exercises

- Prove that \(\lim_{x\to 0} x^2\cos(\frac{1}{x})=0\).
- Prove that \(\lim_{x\to 4} \frac{x-4}{\sqrt{x}-2}=4\).
- Give examples of functions defined for all real numbers which are
- differentiable.
- continuous but not differentiable.
- not continuous.

- Compute the derivative \(f’(x)\) of \(f(x)=3-5x+7x^7\).
- Compute the derivative \(f’(x)\) of \(f(x)=3x^2\tan x\).
- Compute the derivative \(f’(x)\) of \(f(x)=\frac{1-x}{4+x^2}\).
- Compute the derivative \(f’(x)\) of \(f(x)=e^{3x+x^3}\).
- Find all antiderivatives of \(f(x)=2x^3-5x^4\).
- Compute \(\frac{d}{dx}[\int_1^{x^2} \frac{dt}{1+t^2}]\).
- Evaluate \(\int_0^{\pi/2} 3\sin x \, dx \).

Solutions