Calculus II Prerequisite Review
Use these problems to review and assess your readiness for Calculus 2! These cover the essential Precalculus and Calculus I topics that are most likely to come up time and time again in your Calculus II class, including…
Evaluating Limits (including using L’Hospital’s Rule), Limits at Infinity
Differentiation Rules / Evaluating Derivatives
Integration Methods from Calculus 1 (typically, this includes basic antiderivatives and the Substitution Rule)
… and more!
These problems are tricky, because they go deep into the fine details of each of their associated topics. If you are able to answer each of these with confidence, you are likely well-prepared for your Calculus 2 class. If not, take this as an opportunity to review now, so that when these topics come up, you may focus on learning the new material!
Evaluate the following limits. If any of them do not exist, state so explicitly (including whether it approaches $\infty$ or $-\infty$ when appropriate).
$$\text{(a) }\lim_{x \to \infty} x \sin\left(\frac{1}{x}\right)$$ $$\text{(b) }\lim_{x \to -\infty} \frac{\sqrt{9x^2+ 1}}{1-2x}$$ $$\text{(c) }\lim_{x \to 0^{+}} \left(e^x + 3x\right)^{\frac{1}{x}}$$ $$\text{(d) }\lim_{x \to 0} \frac{\int_0^{\arcsin(3x)} \sin^4 t dt}{x^5}$$
Given $f(x) = \ln\left(\frac{x^3\cos (3x)}{e^{x^2}}\right)$, show that $f'(1) = -3-3\tan(3)$. (Hint: Rewrite $f(x)$ before taking a derivative.)
Below is a partial description of a function $f$ whose inputs are the natural numbers and outputs are given. Assuming the pattern holds for all natural numbers, write an expression that correctly evaluates $f(n)$ for all natural numbers $n$.
| $n \mapsto$ | $f(n)$ |
|---|---|
| $1 \mapsto$ | $-\frac{5}{2}$ |
| $2 \mapsto$ | $\frac{8}{4}$ |
| $3 \mapsto$ | $-\frac{11}{8}$ |
| $4 \mapsto$ | $\frac{14}{16}$ |
Simplify the following expressions:
- (a) $\sec(\arctan(x))$ (Your answer should be an expression without any trig/inverse trig functions);
- (b) $\tan(2A)$, given that $\sin(A) = \frac{5}{13}$, $\frac{\pi}{2} < A < \pi$. (Your answer should be a simplified fraction.)
Evaluate the following integrals: (Hint for (b): complete the square!) $$\text{(a) } \int^{\frac{\sqrt{3}}{2}}_{\frac{1}{2}} \frac{\arcsin(x)}{\sqrt{1-x^2}}dx$$ $$\text{(b) } \int \frac{1}{x^2 + 6x+10} dx$$ $$\text{(c) } \int x^3 \sqrt{9-x^2}dx$$
The Squeeze Theorem is stated as follows:
For any function $g(x)$ such that $f(x) \leq g(x) \leq h(x)$ for all $x$ near $a$, if $\lim_{x\to a} f(x) = \lim_{x\to a} h(x) =L$, then $\lim_{x\to a} g(x) = L$.
Use the Squeeze Theorem to evaluate $\lim _{x \to 0} x^3 \cos(\frac{1}{x^3})$. Explicitly state what $f(x)$ and $h(x)$ are for this limit.
- Suppose we wish to approximate the value of $\int_2^7 x dx$ using $R_{100}$, where $R_n$ is the right Riemann sum using $n$ rectangles. Show without using a calculator that $R_{100} = \frac{181}{8}$. You may find the following formula useful:
$$\sum_{k=1}^n k = \frac{n(n+1)}{2}$$
Answers are available here. Please comment below if you have any questions regarding these problems, or contact me if you’d like to discuss further one-on-one!
