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$f(x) = \dfrac{x^2 - 36}{x - 6}$ is not defined at $x = 6$ . What value does $f(x)$ approach as $x \to 6$ ?
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A ball's position is $s(t) = 4t^2 + 9t$ meters after $t$ seconds. Find its average velocity on $[0, 6]$ , in m/s.
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$f(x) = 8x^{ 5 } + 3x$ . Find $f'(x)$ .
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$x^2 + 6y^2 = 63$ . Find $\dfrac{dy}{dx}$ in terms of $x$ and $y$ .
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Find $\displaystyle\lim_{x \to 0} \dfrac{e^{ 5x} - 1}{x}$ .
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A car drives at 30 mph for 2 hours, then at 57 mph until hour 5. How far does it travel, in miles?
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Find $F$ with $F'(x) = 2x\cos(x^2) + 4$ and $F(0) = 0$ .
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Find $F$ with $F'(x) = 8x e^x$ and $F(0) = 0$ .
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Evaluate $\displaystyle\int_1^{\infty} \dfrac{dx}{x^{ 4 } }$ .
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Which function is a solution of $\dfrac{dy}{dx} = 5y$ ?
1. $y = 3e^{ 5x}$
2. $y = e^{3x}$
3. $y = 5xy$
4. $y = e^{x} + 5$
Which function is a solution of $\dfrac{dy}{dx} = 6y$ ?
1. $y = 3e^{ 6x}$
2. $y = e^{3x}$
3. $y = 6xy$
4. $y = e^{x} + 6$
Which function is a solution of $\dfrac{dy}{dx} = 9y$ ?
1. $y = 3e^{ 9x}$
2. $y = e^{3x}$
3. $y = 9xy$
4. $y = e^{x} + 9$
Which function is a solution of $\dfrac{dy}{dx} = 2y$ ?
1. $y = 3e^{ 2x}$
2. $y = e^{3x}$
3. $y = 2xy$
4. $y = e^{x} + 2$
Which function is a solution of $\dfrac{dy}{dx} = 7y$ ?
1. $y = 3e^{ 7x}$
2. $y = e^{3x}$
3. $y = 7xy$
4. $y = e^{x} + 7$
Which function is a solution of $\dfrac{dy}{dx} = 4y$ ?
1. $y = 3e^{ 4x}$
2. $y = e^{3x}$
3. $y = 4xy$
4. $y = e^{x} + 4$
Which function is a solution of $\dfrac{dy}{dx} = 8y$ ?
1. $y = 3e^{ 8x}$
2. $y = e^{3x}$
3. $y = 8xy$
4. $y = e^{x} + 8$
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$\dfrac{dy}{dx} = 1x + y$ and $y(1) = 3$ . Use one step of Euler's method with $h = 0.1$ to estimate $y(1.1)$ .
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A particle has velocity $v(t) = 2t + 7$ and starts at $s(0) = 4$ . Find its position at $t = 5$ .
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$x = t^2$ , $y = t^3 + 9t$ . Find $\dfrac{dy}{dx}$ at $t = 3$ .
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Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 6n}{n + 3}$ converge?
1. No: the terms tend to 6 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 6
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 5n}{n + 8}$ converge?
1. No: the terms tend to 5 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 5
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 5n}{n + 2}$ converge?
1. No: the terms tend to 5 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 5
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 2n}{n + 1}$ converge?
1. No: the terms tend to 2 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 2
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 8n}{n + 4}$ converge?
1. No: the terms tend to 8 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 8
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 3n}{n + 6}$ converge?
1. No: the terms tend to 3 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 3
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 4n}{n + 6}$ converge?
1. No: the terms tend to 4 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 4
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 7n}{n + 8}$ converge?
1. No: the terms tend to 7 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 7
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 6n}{n + 7}$ converge?
1. No: the terms tend to 6 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 6
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 6n}{n + 4}$ converge?
1. No: the terms tend to 6 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 6
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 3n}{n + 7}$ converge?
1. No: the terms tend to 3 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 3
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 5n}{n + 7}$ converge?
1. No: the terms tend to 5 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 5
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 1n}{n + 3}$ converge?
1. No: the terms tend to 1 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 1
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 7n}{n + 6}$ converge?
1. No: the terms tend to 7 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 7
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 5n}{n + 4}$ converge?
1. No: the terms tend to 5 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 5
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 3n}{n + 9}$ converge?
1. No: the terms tend to 3 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 3
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 3n}{n + 8}$ converge?
1. No: the terms tend to 3 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 3
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 9n}{n + 2}$ converge?
1. No: the terms tend to 9 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 9
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 9n}{n + 1}$ converge?
1. No: the terms tend to 9 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 9
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 7n}{n + 5}$ converge?
1. No: the terms tend to 7 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 7
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 8n}{n + 5}$ converge?
1. No: the terms tend to 8 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 8
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 4n}{n + 9}$ converge?
1. No: the terms tend to 4 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 4
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 7n}{n + 7}$ converge?
1. No: the terms tend to 7 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 7
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 2n}{n + 6}$ converge?
1. No: the terms tend to 2 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 2
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 9n}{n + 4}$ converge?
1. No: the terms tend to 9 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 9
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 9n}{n + 6}$ converge?
1. No: the terms tend to 9 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 9
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 7n}{n + 2}$ converge?
1. No: the terms tend to 7 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 7
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 2n}{n + 4}$ converge?
1. No: the terms tend to 2 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 2
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 4n}{n + 1}$ converge?
1. No: the terms tend to 4 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 4
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 2n}{n + 2}$ converge?
1. No: the terms tend to 2 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 2
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 1n}{n + 5}$ converge?
1. No: the terms tend to 1 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 1
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 4n}{n + 7}$ converge?
1. No: the terms tend to 4 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 4
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 3n}{n + 3}$ converge?
1. No: the terms tend to 3 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 3
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 1n}{n + 8}$ converge?
1. No: the terms tend to 1 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 1
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 3n}{n + 4}$ converge?
1. No: the terms tend to 3 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 3
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 6n}{n + 5}$ converge?
1. No: the terms tend to 6 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 6
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 3n}{n + 1}$ converge?
1. No: the terms tend to 3 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 3
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 5n}{n + 5}$ converge?
1. No: the terms tend to 5 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 5
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 2n}{n + 5}$ converge?
1. No: the terms tend to 2 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 2
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 2n}{n + 7}$ converge?
1. No: the terms tend to 2 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 2
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 4n}{n + 2}$ converge?
1. No: the terms tend to 4 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 4
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 8n}{n + 2}$ converge?
1. No: the terms tend to 8 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 8
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 8n}{n + 3}$ converge?
1. No: the terms tend to 8 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 8
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 6n}{n + 8}$ converge?
1. No: the terms tend to 6 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 6
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 6n}{n + 1}$ converge?
1. No: the terms tend to 6 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 6
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 1n}{n + 6}$ converge?
1. No: the terms tend to 1 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 1
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 8n}{n + 7}$ converge?
1. No: the terms tend to 8 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 8
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 5n}{n + 9}$ converge?
1. No: the terms tend to 5 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 5
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 4n}{n + 3}$ converge?
1. No: the terms tend to 4 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 4
3. The nth-term test cannot decide
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{ 8n}{n + 8}$ converge?
1. No: the terms tend to 8 ≠ 0, so it diverges (nth-term test)
2. Yes, because each term is less than 8
3. The nth-term test cannot decide
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Find the degree-2 Maclaurin polynomial of $f(x) = e^{ 8x}$ .
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Find $\displaystyle\lim_{x \to 7} \dfrac{x^2 - 15x + 56}{x - 7}$ .
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$f(x) = x^2 + 4x$ . Use $f'(4) = \displaystyle\lim_{h \to 0} \dfrac{f(4 + h) - f(4)}{h}$ to find $f'(4)$ .
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$f(x) = 8e^x + 5x^2$ . Find $f'(x)$ .
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Find the slope of the tangent line to the circle $x^2 + y^2 = 25$ at the point $(4, 3)$ .
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$f(x) = x^3 - 12x$ . At which $x$ does $f$ have a local maximum?
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Estimate $\displaystyle\int_0^{ 5 } x^2\,dx$ with a left Riemann sum of 5 subintervals of width 1.
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Evaluate $\displaystyle\int_0^1 2x(x^2 + 1)^{ 4 }\,dx$ .
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Find $F$ with $F'(x) = 5x^2 e^x$ and $F(0) = 0$ .
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Evaluate $\displaystyle\int_0^{ 25 } \dfrac{dx}{\sqrt{x} }$ .
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In the slope field of $\dfrac{dy}{dx} = x + 3y$ , what is the slope of the segment at $(-3, 1)$ ?
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A population follows $\dfrac{dP}{dt} = \dfrac{ 7 }{10}P\left(1 - \dfrac{P}{ 558 }\right)$ . What is its carrying capacity?
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Find the area of the region between $y = 5x$ and $y = x^2$ .
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$x = t^2$ , $y = 8t^3$ for $t > 0$ . Find $\dfrac{d^2y}{dx^2}$ in terms of $t$ .
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On the interval, $|f^{(2)}(x)| \le 2$ . Bound the error of the degree-1 Taylor polynomial centered at $a$ , at a point with $|x - a| = \tfrac{1}{2}$ .
$f(x) = \begin{cases} x + 6 & \text{if } x < 5 \\ 2x & \text{if } x \ge 5 \end{cases}$ . What is $\displaystyle\lim_{x \to 5} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $11$
3. $10$
$f(x) = \begin{cases} x + 7 & \text{if } x < 3 \\ 4x & \text{if } x \ge 3 \end{cases}$ . What is $\displaystyle\lim_{x \to 3} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $10$
3. $12$
$f(x) = \begin{cases} x + 2 & \text{if } x < 5 \\ 4x & \text{if } x \ge 5 \end{cases}$ . What is $\displaystyle\lim_{x \to 5} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $7$
3. $20$
$f(x) = \begin{cases} x + 9 & \text{if } x < 1 \\ 2x & \text{if } x \ge 1 \end{cases}$ . What is $\displaystyle\lim_{x \to 1} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $10$
3. $2$
$f(x) = \begin{cases} x + 1 & \text{if } x < 2 \\ 4x & \text{if } x \ge 2 \end{cases}$ . What is $\displaystyle\lim_{x \to 2} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $3$
3. $8$
$f(x) = \begin{cases} x + 9 & \text{if } x < 6 \\ 4x & \text{if } x \ge 6 \end{cases}$ . What is $\displaystyle\lim_{x \to 6} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $15$
3. $24$
$f(x) = \begin{cases} x + 2 & \text{if } x < 2 \\ 5x & \text{if } x \ge 2 \end{cases}$ . What is $\displaystyle\lim_{x \to 2} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $4$
3. $10$
$f(x) = \begin{cases} x + 9 & \text{if } x < 6 \\ 5x & \text{if } x \ge 6 \end{cases}$ . What is $\displaystyle\lim_{x \to 6} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $15$
3. $30$
$f(x) = \begin{cases} x + 6 & \text{if } x < 3 \\ 2x & \text{if } x \ge 3 \end{cases}$ . What is $\displaystyle\lim_{x \to 3} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $9$
3. $6$
$f(x) = \begin{cases} x + 2 & \text{if } x < 6 \\ 3x & \text{if } x \ge 6 \end{cases}$ . What is $\displaystyle\lim_{x \to 6} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $8$
3. $18$
$f(x) = \begin{cases} x + 7 & \text{if } x < 6 \\ 3x & \text{if } x \ge 6 \end{cases}$ . What is $\displaystyle\lim_{x \to 6} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $13$
3. $18$
$f(x) = \begin{cases} x + 1 & \text{if } x < 4 \\ 3x & \text{if } x \ge 4 \end{cases}$ . What is $\displaystyle\lim_{x \to 4} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $5$
3. $12$
$f(x) = \begin{cases} x + 1 & \text{if } x < 3 \\ 2x & \text{if } x \ge 3 \end{cases}$ . What is $\displaystyle\lim_{x \to 3} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $4$
3. $6$
$f(x) = \begin{cases} x + 9 & \text{if } x < 4 \\ 3x & \text{if } x \ge 4 \end{cases}$ . What is $\displaystyle\lim_{x \to 4} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $13$
3. $12$
$f(x) = \begin{cases} x + 5 & \text{if } x < 4 \\ 2x & \text{if } x \ge 4 \end{cases}$ . What is $\displaystyle\lim_{x \to 4} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $9$
3. $8$
$f(x) = \begin{cases} x + 7 & \text{if } x < 2 \\ 5x & \text{if } x \ge 2 \end{cases}$ . What is $\displaystyle\lim_{x \to 2} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $9$
3. $10$
$f(x) = \begin{cases} x + 1 & \text{if } x < 6 \\ 4x & \text{if } x \ge 6 \end{cases}$ . What is $\displaystyle\lim_{x \to 6} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $7$
3. $24$
$f(x) = \begin{cases} x + 7 & \text{if } x < 4 \\ 3x & \text{if } x \ge 4 \end{cases}$ . What is $\displaystyle\lim_{x \to 4} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $11$
3. $12$
$f(x) = \begin{cases} x + 4 & \text{if } x < 6 \\ 5x & \text{if } x \ge 6 \end{cases}$ . What is $\displaystyle\lim_{x \to 6} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $10$
3. $30$
$f(x) = \begin{cases} x + 4 & \text{if } x < 1 \\ 3x & \text{if } x \ge 1 \end{cases}$ . What is $\displaystyle\lim_{x \to 1} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $5$
3. $3$
$f(x) = \begin{cases} x + 1 & \text{if } x < 3 \\ 3x & \text{if } x \ge 3 \end{cases}$ . What is $\displaystyle\lim_{x \to 3} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $4$
3. $9$
$f(x) = \begin{cases} x + 9 & \text{if } x < 3 \\ 2x & \text{if } x \ge 3 \end{cases}$ . What is $\displaystyle\lim_{x \to 3} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $12$
3. $6$
$f(x) = \begin{cases} x + 7 & \text{if } x < 2 \\ 3x & \text{if } x \ge 2 \end{cases}$ . What is $\displaystyle\lim_{x \to 2} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $9$
3. $6$
$f(x) = \begin{cases} x + 4 & \text{if } x < 6 \\ 2x & \text{if } x \ge 6 \end{cases}$ . What is $\displaystyle\lim_{x \to 6} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $10$
3. $12$
$f(x) = \begin{cases} x + 8 & \text{if } x < 6 \\ 3x & \text{if } x \ge 6 \end{cases}$ . What is $\displaystyle\lim_{x \to 6} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $14$
3. $18$
$f(x) = \begin{cases} x + 9 & \text{if } x < 4 \\ 2x & \text{if } x \ge 4 \end{cases}$ . What is $\displaystyle\lim_{x \to 4} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $13$
3. $8$
$f(x) = \begin{cases} x + 6 & \text{if } x < 5 \\ 4x & \text{if } x \ge 5 \end{cases}$ . What is $\displaystyle\lim_{x \to 5} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $11$
3. $20$
$f(x) = \begin{cases} x + 7 & \text{if } x < 2 \\ 4x & \text{if } x \ge 2 \end{cases}$ . What is $\displaystyle\lim_{x \to 2} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $9$
3. $8$
$f(x) = \begin{cases} x + 2 & \text{if } x < 4 \\ 3x & \text{if } x \ge 4 \end{cases}$ . What is $\displaystyle\lim_{x \to 4} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $6$
3. $12$
$f(x) = \begin{cases} x + 1 & \text{if } x < 1 \\ 3x & \text{if } x \ge 1 \end{cases}$ . What is $\displaystyle\lim_{x \to 1} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $2$
3. $3$
$f(x) = \begin{cases} x + 2 & \text{if } x < 1 \\ 2x & \text{if } x \ge 1 \end{cases}$ . What is $\displaystyle\lim_{x \to 1} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $3$
3. $2$
$f(x) = \begin{cases} x + 2 & \text{if } x < 2 \\ 3x & \text{if } x \ge 2 \end{cases}$ . What is $\displaystyle\lim_{x \to 2} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $4$
3. $6$
$f(x) = \begin{cases} x + 7 & \text{if } x < 1 \\ 5x & \text{if } x \ge 1 \end{cases}$ . What is $\displaystyle\lim_{x \to 1} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $8$
3. $5$
$f(x) = \begin{cases} x + 9 & \text{if } x < 3 \\ 5x & \text{if } x \ge 3 \end{cases}$ . What is $\displaystyle\lim_{x \to 3} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $12$
3. $15$
$f(x) = \begin{cases} x + 3 & \text{if } x < 1 \\ 5x & \text{if } x \ge 1 \end{cases}$ . What is $\displaystyle\lim_{x \to 1} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $4$
3. $5$
$f(x) = \begin{cases} x + 8 & \text{if } x < 2 \\ 2x & \text{if } x \ge 2 \end{cases}$ . What is $\displaystyle\lim_{x \to 2} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $10$
3. $4$
$f(x) = \begin{cases} x + 7 & \text{if } x < 4 \\ 5x & \text{if } x \ge 4 \end{cases}$ . What is $\displaystyle\lim_{x \to 4} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $11$
3. $20$
$f(x) = \begin{cases} x + 8 & \text{if } x < 5 \\ 5x & \text{if } x \ge 5 \end{cases}$ . What is $\displaystyle\lim_{x \to 5} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $13$
3. $25$
$f(x) = \begin{cases} x + 4 & \text{if } x < 1 \\ 2x & \text{if } x \ge 1 \end{cases}$ . What is $\displaystyle\lim_{x \to 1} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $5$
3. $2$
$f(x) = \begin{cases} x + 9 & \text{if } x < 1 \\ 4x & \text{if } x \ge 1 \end{cases}$ . What is $\displaystyle\lim_{x \to 1} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $10$
3. $4$
$f(x) = \begin{cases} x + 5 & \text{if } x < 6 \\ 2x & \text{if } x \ge 6 \end{cases}$ . What is $\displaystyle\lim_{x \to 6} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $11$
3. $12$
$f(x) = \begin{cases} x + 7 & \text{if } x < 1 \\ 4x & \text{if } x \ge 1 \end{cases}$ . What is $\displaystyle\lim_{x \to 1} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $8$
3. $4$
$f(x) = \begin{cases} x + 4 & \text{if } x < 3 \\ 2x & \text{if } x \ge 3 \end{cases}$ . What is $\displaystyle\lim_{x \to 3} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $7$
3. $6$
$f(x) = \begin{cases} x + 5 & \text{if } x < 2 \\ 5x & \text{if } x \ge 2 \end{cases}$ . What is $\displaystyle\lim_{x \to 2} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $7$
3. $10$
$f(x) = \begin{cases} x + 3 & \text{if } x < 5 \\ 3x & \text{if } x \ge 5 \end{cases}$ . What is $\displaystyle\lim_{x \to 5} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $8$
3. $15$
$f(x) = \begin{cases} x + 3 & \text{if } x < 5 \\ 5x & \text{if } x \ge 5 \end{cases}$ . What is $\displaystyle\lim_{x \to 5} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $8$
3. $25$
$f(x) = \begin{cases} x + 1 & \text{if } x < 5 \\ 2x & \text{if } x \ge 5 \end{cases}$ . What is $\displaystyle\lim_{x \to 5} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $6$
3. $10$
$f(x) = \begin{cases} x + 7 & \text{if } x < 5 \\ 3x & \text{if } x \ge 5 \end{cases}$ . What is $\displaystyle\lim_{x \to 5} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $12$
3. $15$
$f(x) = \begin{cases} x + 2 & \text{if } x < 4 \\ 4x & \text{if } x \ge 4 \end{cases}$ . What is $\displaystyle\lim_{x \to 4} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $6$
3. $16$
$f(x) = \begin{cases} x + 9 & \text{if } x < 6 \\ 3x & \text{if } x \ge 6 \end{cases}$ . What is $\displaystyle\lim_{x \to 6} f(x)$ ?
1. It does not exist: the left and right limits differ
2. $15$
3. $18$
For $f(x) = x^2$ , $f'(x) = 2x$ . Draw the tangent line to the graph at $x = 1$ : place two points on it.
$f(x) = 3\sin x + 5\cos x$ . Find $f'(x)$ .
$f(x) = x^3 + 8x$ has an inverse $g$ . Note $f(1) = 9$ . Find $g'(9)$ .
$f(x) = x^3 - 3x^2$ . At which $x$ does $f$ have an inflection point?
Use geometry: find $\displaystyle\int_{-6}^{ 6 } \sqrt{ 36 - x^2}\,dx$ .
Evaluate $\displaystyle\int_0^1 \dfrac{2x}{x^2 + 2}\,dx$ .
$\dfrac{ 8 }{(x - 4)(x - 6)} = \dfrac{A}{x - 4} + \dfrac{B}{x - 6}$ . Find $A$ .
Solve $\dfrac{dy}{dx} = \dfrac{x}{y}$ with $y(0) = 8$ (so $y > 0$ ). Give $y$ in terms of $x$ .
The logistic solution $P(t) = \dfrac{ 203 }{1 + Ae^{-rt} }$ has $P(0) = 58$ . Find $A$ .
Find the area of the region between $x = y^2$ and $x = 9$ , integrating with respect to $y$ .
Find the length of the curve $x = 3\cos t$ , $y = 3\sin t$ , $0 \le t \le 2\pi$ .
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^2 + 7}$ converge?
1. Yes: it is smaller than the convergent ∑ 1/n²
2. No: it is smaller than the divergent ∑ 1/n
3. No test applies
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^2 + 8}$ converge?
1. Yes: it is smaller than the convergent ∑ 1/n²
2. No: it is smaller than the divergent ∑ 1/n
3. No test applies
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^2 + 9}$ converge?
1. Yes: it is smaller than the convergent ∑ 1/n²
2. No: it is smaller than the divergent ∑ 1/n
3. No test applies
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^2 + 2}$ converge?
1. Yes: it is smaller than the convergent ∑ 1/n²
2. No: it is smaller than the divergent ∑ 1/n
3. No test applies
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^2 + 5}$ converge?
1. Yes: it is smaller than the convergent ∑ 1/n²
2. No: it is smaller than the divergent ∑ 1/n
3. No test applies
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^2 + 6}$ converge?
1. Yes: it is smaller than the convergent ∑ 1/n²
2. No: it is smaller than the divergent ∑ 1/n
3. No test applies
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^2 + 1}$ converge?
1. Yes: it is smaller than the convergent ∑ 1/n²
2. No: it is smaller than the divergent ∑ 1/n
3. No test applies
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^2 + 4}$ converge?
1. Yes: it is smaller than the convergent ∑ 1/n²
2. No: it is smaller than the divergent ∑ 1/n
3. No test applies
Does $\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^2 + 3}$ converge?
1. Yes: it is smaller than the convergent ∑ 1/n²
2. No: it is smaller than the divergent ∑ 1/n
3. No test applies
Find the radius of convergence of $\displaystyle\sum_{n=0}^{\infty} \dfrac{x^n}{ 8^n}$ .

Answers

$12$
$33$
$40 x^{4} + 3$
$- \frac{x}{6 y}$
$5$
$231$
$4 x + \sin{\left(x^{2} \right)}$
$8 \left(x - 1\right) e^{x} + 8$
$\frac{1}{3}$
$y = 3e^{ 5x}$
$\frac{17}{5}$
$64$
$6$
No: the terms tend to 6 ≠ 0, so it diverges (nth-term test)
$32 x^{2} + 8 x + 1$
$-1$
$12$
$10 x + 8 e^{x}$
$- \frac{4}{3}$
$-2$
$30$
$\frac{31}{5}$
$5 \left(x^{2} - 2 x + 2\right) e^{x} - 10$
$10$
$0$
$558$
$\frac{125}{6}$
$\frac{6}{t}$
$\frac{1}{4}$
It does not exist: the left and right limits differ
$y = 2 x - 1$
$- 5 \sin{\left(x \right)} + 3 \cos{\left(x \right)}$
$\frac{1}{11}$
$1$
$18 \pi$
$\ln{\left(\frac{3}{2} \right)}$
$-4$
$\sqrt{x^{2} + 64}$
$\frac{5}{2}$
$36$
$6 \pi$
Yes: it is smaller than the convergent ∑ 1/n²
$8$