Math 3A Exam II Practice

These are a collection of problems you can use to practice for the exam. To see the answers go to this page.

P-level

Take \((a_n)\) to be a non-zero null sequence and \((b_n)\) to be a positive null sequence.. Determine the following limits:
1. \(\lim\limits_{n\to \infty}(a_n+2)\)
2. \(\lim\limits_{n\to \infty}(a_n)^2\)
3. \(\lim\limits_{n\to \infty}\frac{1}{|a_n|}\)
4. \(\lim\limits_{n\to \infty}\frac{1}{a_n}\)
5. \(\lim\limits_{n\to \infty}\frac{1}{b_n}\)
6. \(\lim\limits_{n\to \infty}2^\frac{1}{a_n}\)
7. \(\lim\limits_{n\to \infty}\sin(a_n)\)
8. \(\lim\limits_{n\to \infty}\cos(a_n)\)
9. \(\lim\limits_{n\to \infty}\tan(a_n)\)
10. \(\lim\limits_{n\to\infty}\frac{\sin(a_n)}{a_n}\)
11. \(\lim\limits_{n\to\infty}\frac{\cos(a_n)}{a_n}\)
12. \(\lim\limits_{n\to\infty}\frac{\cos(b_n)}{b_n}\)
13. \(\lim\limits_{n\to\infty}\frac{\tan(a_n)}{a_n}\)
14. \(\lim\limits_{n\to\infty}\frac{1-\cos(a_n)}{a_n}\)
15. \(\lim\limits_{n\to\infty}\frac{\cos(a_n)-1}{a_n}\)
16. \(\lim\limits_{n\to\infty}\frac{e^{a_n}-1}{a_n}\)
17. \(\lim\limits_{n\to\infty}\frac{1-e^{a_n}}{a_n}\)
18. \(\lim\limits_{n\to\infty}\arctan(a_n)\)
Take \((a_n)\), \((b_n)\), and \((c_n)\) to be sequences that converge to \(2\), \(3\) and \(5\) respectively. Determine the following limits:
1. \(\lim\limits_{n\to\infty}\frac{a_n-2}{b_n}\)
2. \(\lim\limits_{n\to\infty}\frac{a_nb_n}{c_n+5}\)
3. \(\lim\limits_{n\to\infty}\frac{\sqrt{a_n+2}}{c_n+\mathrm{e}^2}\)
4. \(\lim\limits_{n\to\infty}\left(\arctan(c_n)+\ln(b_n)\right)\)
Take \(p_1=(-1,3),p_2=(4,6),p_3=(5,8)\). Use the shoelace formula to determine the area of the triangle \(\Delta p_1p_2p_3\) and to determine whether or not the triangle is positively or negatively oriented.
Take \(p_1=(-1,3),p_2=(4,6),p_3=(5,8)\). Use Heron’s Formula to calculate the area of the triangle \(\Delta p_1p_2p_3\).
Take \(D\) to be the disk of radius \(8\) with center at \((5,3)\). Find a path that parameterizes the boundary of \(D\) in the clockwise direction.
Determine whether the sequence diverges to either infinity or negative infinity:
1. \(a_n = 8^{n}\)
2. \(a_n = \ln(n^2+1)\)
3. \(a_n = -n^6\)
4. \(a_n = \dfrac{-n^3+5n}{n^2+n-1}\)
Identify an example of a sequence \((a_n)\) with the following properties:
1. \((a_n)\) is strictly increasing and converges to \(8\)
2. \((a_n)\) is strictly decreasing and converges to \(4\)
Take \(f\) to be a function with the property that \[\lim_{x\to 2^-} f(x)= 5 = \lim_{x\to 2^+} f(x).\] Determine the value of \(f(2)\) so that \(f\) is continuous at \(2\).
Take \(f\) to be a function with the property that \[\lim_{x\to 2} f(x)=4\quad\text{and}\quad f(2)=1.\] Determine if \(f\) is continuous at \(2\).
Take \(f\) to be a function defined on \(\mathbb{R}\) so that \(\displaystyle\lim_{x\to 2}f(x)=9\).
1. Determine \(\displaystyle\lim_{x\to 2^-}f(x).\)
2. Determine a requirement on \(f\) so that \(f\) is continuous at \(2\).
Give a sketch of a function \(f\) that is continuous at \(x=1\), not continuous at \(x=2\), and diverges to negative infinity at \(x=4.\)
Take \(f\) to be the function that is given by \[f(x) = \frac{x^2-16}{x-4}.\] Determine the maximal domain of \(f\) and a continuous extension of \(f\) to all of \(\mathbb R\).
For each function \(f\) that is given below, classify all zeros of \(f\) by finding the largest natural number \(n\) and a real number \(x_0\) so that \(f\) is \(\LO((x-x_0)^n).\)
1. \(f(x) = (x-6)^3(x-5)^2(x+2)^4\)
2. \(f(x) = \dfrac{x^3(x-4)^6(x+5)^3}{(x+7)(x+9)}\)
For each function \(f\) and \(x_0\) that is given below, find the largest natural numbers \(m\) and \(n\) so that \(f\) is \(\LO((x-x_0)^m)\) and \(\Lo((x-x_0)^n).\)
1. \(f(x) = (x-2)(x-6)^2(x-8)^3\) and \(x_0=6\)
2. \(f(x) = (x-5)\sin((x-3)^2)\) and \(x_0=3\)
3. \(f(x) = x(1-\cos(x+6))\) and \(x_0=-6\)
4. \(f(x) = 1-\mathrm{e}^{x}\) and \(x_0=0\)
For each function \(f\) that is given below, find the largest natural numbers \(m\) so that \(f\) is \(\Lo((x-x_0)^m)\):
1. \(x_0=-1\) and \(f(x)=(x+1)^{33}(x+8)^{20}(x-1)^{17}\);
2. \(x_0=2\) and \(f(x)=(x-4)^{14}\sin^{16}(x-2)\);
3. \(x_0=1\) and \(f(x)=(x+1)^{3}(1-\cos(x-1))^{10}\);
For each function \(f\) that is given below, find the largest natural numbers \(m\) so that \(f\) is \(\LO((x-x_0)^m)\):
1. \(x_0=-6\) and \(f(x)=(x-6)^{24}(x+3)^{25}(x+6)^{45}\);
2. \(x_0=7\) and \(f(x)=(x-7)^{14}\sin^{13}(x-3)\);
3. \(x_0=5\) and \(f(x)=(x-2)^{53}(1-\cos(x-5))^{2}\);
4. \(x_0=3\) and \(f(x)=(x-4)^{14}\sin^{23}(x-3)\);
Take \(P\) to be a partition of \([1,5]\) with domain \(\{0,1,2,3,4\}\) and \(P=\left(1,\tfrac{5}{2},\frac{11}{4},\frac{22}{5},5\right)\). Determine the mesh of \(P\).
Construct an even partition \(P\) of \([2,5]\) that has five intervals.
Identify a midpoint tagging \(\tau\) of the partition \(P\) of \([2,5]\) that has three intervals and \(P=(2,3,4.5,5).\)
Calculate the following by interpreting the definite integral as signed area.
1. \(\displaystyle\int_{-1}^{4}-2\,\mathrm{d}x\)
2. \(\displaystyle\int_{-6}^{6}\sqrt{36-x^2}\,\mathrm{d}x\)
3. \(\displaystyle\int_{-2}^{1}3x\,\mathrm{d}x\)
Determine the following limits
1. \(\lim\limits_{x\to 1}(4x^2+2x)\)
2. \(\lim\limits_{x\to 2}|x-2|\)
3. \(\lim\limits_{x\to 5}\sin(x)\)
4. \(\lim\limits_{x\to \pi}\tan(x)\)
5. \(\lim\limits_{x\to 3\pi}\sin(x)\)
6. \(\lim\limits_{x\to \pi}\mathrm{e}^x\)
7. \(\lim\limits_{x\to \pi}(\mathrm{e}^4+\ln(5))\)
8. \(\lim\limits_{x\to 0}\frac{\mathrm{e}^{x}-1}{x}\)
9. \(\lim\limits_{x\to 1}\frac{\mathrm{e}^{x}-1}{x}\)
10. \(\lim\limits_{x\to 2}20\)
11. \(\lim\limits_{x\to 1}\frac{x^2-2x}{x+2}\)
12. \(\lim\limits_{x\to 0^+}\csc(x)\)
13. \(\lim\limits_{x\to 0^-}\csc(x)\)
14. \(\lim\limits_{x\to 1^+}\frac{x-2}{x-1}\)
15. \(\lim\limits_{x\to 1^-}\frac{x-2}{x-1}\)
16. \(\lim\limits_{x\to 2}\frac{1-\cos(x-2)}{x-2}\)
17. \(\lim\limits_{x\to \frac{\pi}{2}^-}\tan(x)\)
18. \(\lim\limits_{x\to \frac{\pi}{2}^+}\tan(x)\)
19. \(\lim\limits_{x\to \frac{\pi}{2}^+}\csc(x)\)
20. \(\lim\limits_{x\to \frac{\pi}{2}^-}\sec(x)\)
21. \(\lim\limits_{x\to 2}|x|\)
22. \(\lim\limits_{x\to \mathrm{e}}\ln(x^2)\)
23. \(\lim\limits_{x\to 5}\log_2(2x+6)\)
24. \(\lim\limits_{x\to 0^+}\log_2(2x)\)
25. \(\lim\limits_{x\to 0^+}\log_\frac{1}{2}(x)\)
Determine the following limits
1. \(\lim\limits_{x\to \infty}(4x^2+2x)\)
2. \(\lim\limits_{x\to \infty}(-4x^2+2x)\)
3. \(\lim\limits_{x\to \infty}|x-2|\)
4. \(\lim\limits_{x\to \infty}\sin(x)\)
5. \(\lim\limits_{x\to \infty}\tan(x)\)
6. \(\lim\limits_{x\to \infty}\cos(x)\)
7. \(\lim\limits_{x\to \infty}\mathrm{e}^x\)
8. \(\lim\limits_{x\to \infty} 2^x\)
9. \(\lim\limits_{x\to \infty}\left(\tfrac{1}{2}\right)^x\)
10. \(\lim\limits_{x\to \infty}\ln(x)\)
11. \(\lim\limits_{x\to \infty}\log_\frac{1}{2}(x)\)
12. \(\lim\limits_{x\to \infty}\arctan(x)\)
13. \(\lim\limits_{x\to \infty}\sqrt{x}\)
14. \(\lim\limits_{x\to -\infty}(4x^2+2x)\)
15. \(\lim\limits_{x\to -\infty}|x-2|\)
16. \(\lim\limits_{x\to -\infty}\sin(x)\)
17. \(\lim\limits_{x\to -\infty}\tan(x)\)
18. \(\lim\limits_{x\to -\infty}\cos(x)\)
19. \(\lim\limits_{x\to -\infty}\mathrm{e}^x\)
20. \(\lim\limits_{x\to -\infty}2^x\)
21. \(\lim\limits_{x\to -\infty}\left(\tfrac{1}{2}\right)^x\)
22. \(\lim\limits_{x\to -\infty}\arctan(x)\)
Take \(a_n\), \(b_n\) and \(c_n\) to be convergent sequences that converge to \(2\), \(-5\), and \(10\), respectively. Compute the following limit: \[\lim_{n\to\infty}\left(\frac{\sqrt{a_n}+3|b_n|}{(c_n)^3+1}+\frac{n+3}{a_nn+1}\right).\]
Compute the following limits.
1. \(\displaystyle \lim_{n\to\infty}\frac{n^2+n+1}{3n^2+n+1}\)
2. \(\displaystyle \lim_{x\to\infty}\frac{\mathrm{e}^x+1}{2\mathrm{e}^x+3}\)
3. \(\displaystyle\lim_{x\to 3}\frac{\sin(x-3)}{x-3}\)
4. \(\displaystyle \lim_{x\to 5}\frac{x+4}{\mathrm{e}^{x-5}}\)
5. \(\displaystyle \lim_{n\to\infty}\frac{4n^3+n^2+1}{-10n^4+1}\)
6. \(\displaystyle \lim_{x\to\infty}\frac{9x^2-1}{x^2+3}\)
7. \(\displaystyle \lim_{x\to 10}\frac{x-10}{\cos(x-10)}\)
8. \(\displaystyle\lim_{x\to 3}\frac{1-\cos(x-3)}{x-3}\)
Determine the following limits:
1. \(\displaystyle \lim_{x\to 2}\frac{\sin(x-2)}{x+2}\)
2. \(\displaystyle \lim_{x\to 2}\frac{\sin(x-2)}{x-2}\)
3. \(\displaystyle \lim_{x\to 3}\frac{1-\cos(x-2)}{x+3}\)
4. \(\displaystyle \lim_{x\to 4}\frac{1-\cos(x-4)}{x-4}\)
5. \(\displaystyle \lim_{x\to 0}\frac{\mathrm{e}^{x}-1}{x}\)
6. \(\displaystyle \lim_{x\to 1}\frac{\mathrm{e}^{x}-1}{x}\)
7. \(\displaystyle \lim_{x\to 2}\frac{\mathrm{e}^{x}-\mathrm{e}^2}{x-2}\)
Take \(f\) to be a continuous function on \((-\infty,3)\cup(3,\infty)\) such that \[\lim_{x\to 3}f(x)=1,\quad \lim_{x\to \infty}f(x)=9\quad \text{and}\quad \lim_{x\to -\infty}f(x)=\infty.\] Determine the vertical and horizontal asymptotes of \(f\).
Take \(f\) to be a continuous function on \((-\infty,5)\cup(5,7)\cup(7,\infty)\) such that \[\lim_{x\to 5^+}f(x)=1,\quad \lim_{x\to 5^-}f(x)=9,\quad \lim_{x\to 7^-}f(x)=1,\quad \lim_{x\to 7^+}f(x)=-\infty,\quad \lim_{x\to \infty}f(x)=2,\quad \text{and}\quad \lim_{x\to -\infty}f(x)=\mathrm{e}^2.\] Determine the vertical and horizontal asymptotes of \(f\).

C-level

Take \(a_n\) to be the sequence that is given by \[a_n = \tfrac{n^2 -20n\sin(n)}{10n^2+n-1}.\] Calculate \(\displaystyle\lim_{n\to \infty} a_n\) and be sure to carefully justify your reasoning.
Calculate \(\displaystyle\lim_{n\to \infty} n\left(\sqrt{36+\tfrac{1}{n}} - 6 \right)\) and be sure to carefully justify your reasoning.
Suppose that \((\theta_n)\) is a null sequence with only nonzero terms. Calculate the following limits:
1. \(\lim\limits_{n\to\infty} \tfrac{\sin(2\theta_n)}{5\theta_n}.\)
2. \(\lim\limits_{n\to\infty}\tfrac{1-\cos(3\theta_n)}{5\theta_n}.\)
3. \(\lim\limits_{n\to\infty} \tfrac{1-\cos(3\theta_n)}{5\theta_n^2}.\)
4. \(\lim\limits_{n\to\infty} \tfrac{\tan(2\theta_n)}{6\theta_n}.\)
5. \(\lim\limits_{n\to\infty} \tfrac{2\theta_n}{\sin(6\theta_n)}.\)
Take \(f\) to be the function given by \[f(x) =\begin{cases}\dfrac{x^2 + 5 x - 14}{x-2} &\text{if }x\ne 2\\10&\text{if }x = 2.\end{cases}\] Determine \(\lim\limits_{x\to 2}f(x)\) and whether \(f\) is continuous at \(2\).
Determine the way in which the following limits diverge:
1. \(\lim\limits_{x\to 3^+} \tfrac{1}{(x-3)^5}\)
2. \(\lim\limits_{x\to 3^-} \tfrac{1}{(x-3)^5}\)
3. \(\lim\limits_{x\to 3} \tfrac{1}{(x-3)^5}\)
4. \(\lim\limits_{x\to 3^+} \tfrac{1}{(x-3)^2}\)
5. \(\lim\limits_{x\to 3^-} \tfrac{1}{(x-3)^2}\)
6. \(\lim\limits_{x\to 3} \tfrac{1}{(x-3)^2}\)
7. \(\lim\limits_{x\to 3} \tfrac{x-4}{(x-3)^2}\)
8. \(\lim\limits_{x\to 3^+} \tfrac{x-4}{(x-3)^5}\)
9. \(\lim\limits_{x\to 3^-} \tfrac{x-4}{(x-3)^5}\)
10. \(\lim\limits_{x\to 3^+} \sin\left(\tfrac{1}{x-3}\right)\)
11. \(\lim\limits_{x\to 3^-} \sin\left(\tfrac{1}{x-3}\right)\)
12. \(\lim\limits_{x\to 3^-} \tan\left(\tfrac{1}{x-3}\right)\)
13. \(\lim\limits_{x\to 3^+} \tan\left(\tfrac{1}{x-3}\right)\)
For each function \(f\) and \(x_0\) that is given below, find the largest natural numbers \(m\) and \(n\) so that \(f\) is \(\LO((x-x_0)^m)\) and \(\Lo((x-x_0)^n).\)
1. \(f(x) = (x-3)\sin((x-3)^2)\) and \(x_0=3\)
2. \(f(x) = (x+6)(1-\cos(x+6))\) and \(x_0=-6\)
3. \(f(x) = x(1-\mathrm{e}^{x})\) and \(x_0=0\)
Determine the following limit and carefully justify your reasoning: \[\displaystyle \lim_{x\to 4}\frac{\sin(5(x-4))}{6(x-4)}.\]
Take \(f\) to be a function with \(\mathcal{D}(f)=(-\infty,4)\cup(4,\infty)\) and \[x^2+x-20\leq f(x) \leq (x-4)(5^{x-3}+4)\] for \(x\in(4,\infty).\) Use the squeeze theorem to determine the limit: \[\displaystyle\lim_{x\to 4^+}\frac{f(x)}{x-4}.\]
Take \(f\) to be a function with \(\mathcal{D}(f)=(-\infty,0)\cup(0,\infty)\) and \[-\frac{1}{x^3-2x}\leq f(x) \leq -\frac{1}{x2^x+3x}\] for \(x\in(-\sqrt{2},0).\) Use the squeeze theorem to determine the limit: \[\displaystyle\lim_{x\to 0^-}xf(x).\]
Take \(f\) to be a function with \(\mathcal{D}(f)=(-\infty,-5)\cup(-5,\infty)\) and \[\frac{\sin\left(-\frac{\pi}{20}x\right)}{x^2+10x+25}\leq f(x) \leq \frac{\sqrt{2}\cdot 2^{x+5}}{2x^2+20x+50}\] for \(x\in\mathcal{D}(f).\) Use the squeeze theorem to determine the limit: \[\displaystyle\lim_{x\to -5}(x+5)^2f(x).\]
Take \(f\) to be a function with \(\mathcal{D}(f)=(-\infty,6)\cup(6,\infty)\) and \[(x-6)(x^2+2)\leq f(x) \leq (x-6)(6^{8-x}+2)\] for \(x\in(6,\infty).\) Use the squeeze theorem to determine the limit: \[\displaystyle\lim_{x\to 6^+}\frac{f(x)}{x-6}.\]
Calculate the following. If you use any limit law or theorem, make note of it.
1. \(\displaystyle \lim_{n\to\infty}\left[ n\left(\sqrt{144+\frac{1}{n}}-12\right)\right]\)
2. \(\displaystyle \lim_{n\to\infty}\left(\frac{15n^4+200n+1}{16n^4+5n+4}+n\left(\sqrt{144+\frac{1}{n}}-12\right)\right)\)
3. \(\displaystyle\lim_{x\to 3}\sin\left(\frac{x^2 -x - 6}{x-3}\right)\)
4. \(\displaystyle\lim_{x\to 4}\left(\sin\left(\frac{x-4}{x+4}\right)\cos\left(4x\right)+\ln(2)\right)\)
5. \(\displaystyle\lim_{x\to\infty}\frac{-x^2+1}{x+4}\)
6. \(\displaystyle\lim_{x\to -\infty}\frac{-x^2+1}{x+4}\)
7. \(\displaystyle\lim_{x\to -\infty}\frac{\mathrm{e}^x+4}{\arctan(x)}\)
Calculate the following. If you use any limit law or theorem, make note of it.
1. \(\displaystyle \lim_{n\to\infty} \frac{\frac{1}{n}}{\sqrt{196+\frac{1}{n}}-14}\)
2. \(\displaystyle \lim_{n\to\infty}\left(\frac{\mathrm{e}^{-n}+3}{16+\frac{1}{n}-\arctan(n)}+ \frac{\frac{1}{n}}{\sqrt{196+\frac{1}{n}}-14}\right)\)
3. \(\displaystyle\lim_{x\to 1}\exp\left((x-1)\sin\left(\tfrac{1}{x-1}\right)+2\right)\)
4. \(\displaystyle\lim_{x\to 3}\left(30\ln\left(\frac{x-2}{x+3}\right)+3^{\pi}\right)\)
5. \(\displaystyle\lim_{x\to\infty}\frac{x^2+1}{x^3+4}\)
6. \(\displaystyle\lim_{x\to -\infty}\frac{-x^3+x+1}{x^3+4}\)
Take \(f\) to be the function given by \[f(x)=\begin{cases} \dfrac{x^2 + 3 x - 28}{x-4}&\text{ if }x<4\\ 2^{x}-5 &\text{ if }x\geq 4. \end{cases}\]
1. Determine \(\lim\limits_{x\to 4^-}f(x).\)
2. Determine \(\lim\limits_{x\to 4^+}f(x).\)
3. Determine whether \(\lim\limits_{x\to 4}f(x)\) exists and determine the limit if it exists.
Show that the function \(f\) is continuous at \(0\), where \(f\) is given by \[f(x) = \begin{cases}\dfrac{\sin(3x)}{2x} &\text{if } x\ne 0\\\log_2(x+2)+\frac{1}{2}&\text{if } x= 0.\end{cases}\]
Write the function \(f\) as a composite function to determine \(\lim\limits_{x\to 4}f(x)\), where \(f\) is given by \[f(x) = \log_3\left(\frac{9\sin(x-4)}{x-4}\right).\]
Take \(f\) to be the polynomial that is given by \[f(x) = x^7 + 3x- 2.\] Use the intermediate value theorem to show that \(f\) has at least one real root.
For each function \(f\) and each \(x_0\) that is given below, use the difference quotient \(\frac{1}{h}\Delta_{x_0}f(h)\) or \(f^{\text{ave}}_{x_0}(h)\) to determine the derivative of \(f\) at \(x_0\).
1. \(f(x) = 3x^2-2x+2\), \(x_0 = 3\)
2. \(f(x) = \frac{x^2 -x+1}{x}\), \(x_0 = 2\)
3. \(f(x) = \cos(2x)\), \(x_0 = \frac{\pi}{4}\)
4. \(f(x) = \sqrt{x-1}\), \(x_0=17\)
5. \(f(x)= \exp(3x)\), \(x_0=2\)
For each function \(f\) given below, determine the difference quotient \(\frac{1}{h}\Delta_{a}f(x)\) for any \(a\) in the domain of the function and use it to determine its derivative.
1. \(f(x)=\sin(2x+1)\)
2. \(f(x)=\cos(\mathrm{e}x^2)\)
3. \(f(x)=\tan(4x+1)\)
4. \(f(x)=\mathrm{e}^{5x^2+x}\)
5. \(f(x)=5x^2+10x-4\)
Take \(c\) to be the path that is given for each \(t\) by \[c(t)=(t^3-t+3,2\sin(3t)).\] Use limits to directly calculate \(c'(t).\)
Calculate the area of each region \(R\) formed by bounding the following.
1. \(y=4\), \(x=-3\), \(x=10\) and the \(x\)-axis
2. \(y=x-2\), \(x=-2\), \(x=6\) and the \(x\)-axis
3. \(y=\sqrt{100-x^2}\), \(x=-10\), \(x=10\) and the \(x\)-axis
Take \(R\) to be the region formed by bounding \(y=2x^2+1\), \(x\)-axis, \(x=1\), and \(x=4.\) Approximate \(R\) by using three left-endpoint rectangles, three right-endpoint rectangles, and three midpoint rectangles.
Take \(R\) to be the region bounded by \(f(x)=x^2+3,\) \(x=0,\) \(x=2,\) and the \(x\)-axis. Approximate \(R\) by using four right-endpoint rectangles, four left-endpoint rectangles, and four midpoint rectangles. Draw your approximations.
Take \(R\) to be the region bounded by \(f(x)=x^2-3x\), \(x=3\), \(x=5\), and the \(x\)-axis. Approximate \(R\) by using four right-endpoint rectangles, four left-endpoint rectangles, and four midpoint rectangles. Draw your approximations.
Take \(f\) to be the function given by \(f(x)=x^2+2\) and the interval \(I\) to be the interval \(I=[1,4].\) Determine an even partition \(P\) of \(I\) with three intervals and a left tagging \(\tau\) for \(P\) and calculate the quantity \(\mathcal{R}(f,P,\tau).\)
Take \(f\) and \(g\) to be integrable functions with \[\displaystyle\int^{3}_{1}f(x)\,\mathrm{d}x=2\quad\text{and}\quad \int^{3}_{1}g(x)\,\mathrm{d}x=5.\] Calculate the following: \[\int_{1}^{3}(f(x)-5g(x)+1)\,\mathrm{d}x.\]
Take \(f\) to be an integrable function such that \[\int_1^4 f(x)\,\mathrm{d}x=-2\quad\text{and}\quad\int_4^8f(x)\,\mathrm{d}x=9.\] Determine the following definite integral \[\int_{1}^8(f(x)+3)\,\mathrm{d}x.\]
Take \(f\) to be an integrable function such that \[\int_{-1}^3 f(x)\,\mathrm{d}x=4\quad\text{and}\quad\int_{-1}^6f(x)\,\mathrm{d}x=10.\] Determine the following definite integral \[\int_{3}^{6}2f(x)\,\mathrm{d}x.\]
Take \(f\) to be an integrable function with \[2\leq f(x)\leq 5\] on the interval \([1,3].\) Determine an upper and lower bound for the following:
1. \(\displaystyle\int_{2}^{3}f(x)\,\mathrm{d}x\)
2. \(\displaystyle\int_{1}^{3}f(x)\,\mathrm{d}x\)
3. \(\displaystyle\int_{1}^{3}7f(x)\,\mathrm{d}x\)
4. \(\displaystyle\int_{1}^{3}(f(x)+3)\,\mathrm{d}x\)
Use \(\displaystyle\int_0^b x^n\,\mathrm{d}x=\tfrac{1}{n+1}b^{n+1}\) and symmetry to compute the following:
1. \(\displaystyle\int_{-2}^{3}x^7\,\mathrm{d}x\)
2. \(\displaystyle\int_{-3}^{5}x^2\,\mathrm{d}x\)
Take \[f(x)=\begin{cases}4x+2&\text{ if }x<1\\\cos(x-1)+a &\text{ if }x\geq 1,\end{cases}\] where \(a\) is a real number.
1. Calculate \(\displaystyle\lim_{x\to 1^-}f(x).\)
2. Calculate \(\displaystyle\lim_{x\to 1^+}f(x).\)
3. Determine \(a\) so that \(\displaystyle\lim_{x\to 1}f(x)\) exists.
Take \[f(x)=\begin{cases}\mathrm{e}^{x-1}+3&\text{ if }x<1\\5x+a &\text{ if }x\geq 1\end{cases},\] where \(a\) is a real number.
1. Calculate \(\displaystyle\lim_{x\to 1^-}f(x).\)
2. Calculate \(\displaystyle\lim_{x\to 1^+}f(x).\)
3. Determine \(a\) so that \(\displaystyle\lim_{x\to 1}f(x)\) exists.
Take \[f(x)=\begin{cases}10x+3&\text{ if }x<2\\5x+a &\text{ if }x> 2\end{cases},\] where \(a\) is a real number.
1. Calculate \(\displaystyle\lim_{x\to 2^-}f(x).\)
2. Calculate \(\displaystyle\lim_{x\to 2^+}f(x).\)
3. Determine \(a\) so that \(\displaystyle\lim_{x\to 2}f(x)\) exists.
4. Based on your answer to part \(c\), determine if \(f\) is continuous at \(x=2.\)
Take \[f(x)=\begin{cases}5x+1&\text{ if }x<1\\ x+a &\text{ if }1\leq x<b \\x^2+2&\text{if }x\geq b,\end{cases}\] where \(a\) and \(b\) are real numbers. Determine \(a\) and \(b\) so that \(\displaystyle\lim_{x\to 1}f(x)\) and \(\displaystyle\lim_{x\to b}f(x)\) exist.

B-level

Take \(R\) to be the rectangle with ordered vertex set \(((1,2), (5,5), (-1,13),(-5,10))\).
1. Find a translation and rotation so that \((1,2)\) is moved to \((0,0)\) and the edge \(\overline{(1,2)(5,5)}\) of \(R\) is on the \(x\)-axis.
2. Use part a to determine if \((4,5)\) is inside \(R\) or outside \(R\).
Create a function \(f\) that is continuous on \((-\infty,2)\cup(2,3)\cup(3,\infty)\), \(\lim\limits_{x\to 2^+}f(x)\) and \(\lim\limits_{x\to 2^-}f(x)\) exist, but \(\lim\limits_{x\to 2}f(x)\) does not exist and \(\lim\limits_{x\to 3}f(x)\) diverges to infinity.
Calculate the following \[\lim\limits_{x \to \infty}\left(\sqrt{36x^2+4x+1}-6x\right).\]
Determine all horizontal and vertical asymptotes of the function \(f\) that is given by \[f(x) = \begin{cases} \dfrac{5x+1}{3x+1}&\text{ if }x<-\frac{1}{3}\\ \dfrac{5x+1}{(x+3)(x-3)} &\text{ if }-\frac{1}{3}\leq x<3\\ \arctan(x)+\frac{2x}{x+2} &\text{ if }x\geq 3.\end{cases}\]
Construct a continuous path \(c\) with domain \([0,\infty)\) that describes the position of a particle that moves to the right on the line segment from \((-2,3)\) to \((5,7)\), is at \((-2,3)\) at time \(0\), is never at the same point at different time points, that never reaches \((5,7),\) but that has the property that \[\lim_{t\to\infty}\|(5,7)-c(t)\|=0.\]
Construct a continuous path \(c\) with domain \([0,\infty)\) that describes the position of a particle that moves to the right on the line segment from \((-2,3)\) to \((5,7)\), is at \((-2,3)\) at time \(0\), is never at the same point at different time points, that never reaches \((5,7),\) but that has the property that \[\lim_{t\to\infty}\|(5,7)-c(t)\|=\tfrac{1}{2}.\]
Use the bisection method with on \([4,5]\) to approximate a solution to the following equation \[x^2 = 17\] to within an error of no greater than \(\frac{1}{10}\).
Take \(f\) to be the function that is given by \[f(x) = \sin(2x^2+1).\] Show that \(f\) is differentiable using limits for each \(x\) in \(\mathbb R\) and determine \(f^\prime(x)\).
Take \(f\) to be the function that is given by \[f(x) = \mathrm{e}^{2x^2+5x}.\] Show that \(f\) is differentiable using limits for each \(x\) in \(\mathbb R\) and determine \(f^\prime(x)\).
Calculate using Riemann sums the area of the region \(R\) formed by bounding \(y=3x^2+1\), \(x=1\), \(x=4\), and the \(x\)-axis.
Take \(f\) and \(g\) to be functions whose sketches are given below. In addition, the function \(g\) is invertible and has an inverse \(g^{-1}\).
1. \(\lim\limits_{x\to 3}(f(x)+g(x))\)
2. \(\lim\limits_{x\to 1}(f(2x)+g(x-1))\)
3. \(\lim\limits_{x\to\infty}f(x)g(x)\)
4. \(\lim\limits_{x\to 3^+}\frac{1}{f(x)}\)
5. \(\lim\limits_{x\to 3^-}\frac{1}{f(x)}\)
6. \(\lim\limits_{x\to \infty}g(x)\)
7. \(\lim\limits_{x\to -\infty}g(x)\)
8. \(\lim\limits_{x\to 0.1}g^{-1}(x)\)
9. \(\lim\limits_{x\to -0.5}g^{-1}(x)\)
10. \(\lim\limits_{x\to 0.5^-}g^{-1}(x)\)
11. \(\lim\limits_{x\to -1^+}g^{-1}(x)\)
12. \(\lim\limits_{x\to -\infty}(g\circ f)(x)\)

A-level

Identify a path that describes the position in time of a particle that moves along the line segment \(L\) with endpoints \((5,4)\) and \((9,2)\), has domain equal to \(\mathbb{R}\), is at the midpoint of \(L\) at time \(0\), moves only to the left, and reaches all points of \(L\) except the endpoints of \(L\).
Take \(f\) to be the function given by \[f(x)=\begin{cases}5x+1&\text{if }x<1\\ 4&\text{if } x=1\\ 6\ln(\mathrm{e}+x-1)&\text{if }x>1\end{cases}\] and \(g\) to be the piecewise linear function whose sketch is given below.
1. Compute \(\lim\limits_{x\to -1}(f\circ g)(x)\).
2. Compute \(f\left(\lim\limits_{x\to -1}g(x)\right)\).
3. Determine whether the limits in (a) and (b) are the same. If the limits are not the same, explain why.
Take \(f\) to be the function whose sketch is given below. Take \(P\) to be an even partition of the interval \([0,6]\) with three intervals, \(\tau_L\) to be a left endpoint tagging of \(P\), \(\tau_R\) to be a right endpoint tagging of \(P\) and \(\tau_M\) to be a midpoint tagging of \(P\). Compute \(\mathcal{R}(f,P,\tau_L)\), \(\mathcal{R}(f,P,\tau_R)\), and \(\mathcal{R}(f,P,\tau_M).\)
Take \(f\) to be a function defined on \([0,6]\) whose sketch is given below. It is a piecewise function made up of linear functions and a quadratic function. Determine its derivative \(f^\prime\).
Construct a function that is continuous everywhere except at \(3\), that is strictly decreasing to the left of \(3\), is asymptotically equal to \(\left(\tfrac{1}{2}\right)^x\) to the left, is strictly increasing to the right of \(3\), is right continuous at \(3\), is asymptotically equal to \(10x^4\), and has the property that \[\lim_{x\to 3^-}f(x)=5\quad\text{and}\quad \lim_{x\to 3^+}f(x)=7.\]
A polygon \(P\) in the plane is given below. Determine its area.
Take \(R\) to be the region formed by bounding a piecewise function \(f\), the \(x\)-axis, \(x=0\) and \(x=6\). A sketch of \(R\) is given below. The function \(f\) is made up of a linear function and a quadratic function. Set up, but do not evaluate, an integral that equals the area of \(R\) or integrals that equal the area of \(R\).