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