Part 0: Calculus 1 Review

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

Review Exercises

1. Prove that \(\lim_{x\to 0} x^2\cos(\frac{1}{x})=0\).
2. Prove that \(\lim_{x\to 4} \frac{x-4}{\sqrt{x}-2}=4\).
3. Give examples of functions defined for all real numbers which are
   1. differentiable.
   2. continuous but not differentiable.
   3. not continuous.
4. Compute the derivative \(f’(x)\) of \(f(x)=3-5x+7x^7\).
5. Compute the derivative \(f’(x)\) of \(f(x)=3x^2\tan x\).
6. Compute the derivative \(f’(x)\) of \(f(x)=\frac{1-x}{4+x^2}\).
7. Compute the derivative \(f’(x)\) of \(f(x)=e^{3x+x^3}\).
8. Find all antiderivatives of \(f(x)=2x^3-5x^4\).
9. Compute \(\frac{d}{dx}[\int_1^{x^2} \frac{dt}{1+t^2}]\).
10. Evaluate \(\int_0^{\pi/2} 3\sin x \, dx \).

Solutions