conversations listlengths 2 2 |
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[
{
"from": "human",
"value": "Let \\[f(x) = \\left\\{\n\\begin{array}{cl} ax+3, &\\text{ if }x>2, \\\\\nx-5 &\\text{ if } -2 \\le x \\le 2, \\\\\n2x-b &\\text{ if } x <-2.\n\\end{array}\n\\right.\\]Find $a+b$ if the piecewise function is continuous (which means that its graph can be drawn without lifting you... |
[
{
"from": "human",
"value": "A rectangular band formation is a formation with $m$ band members in each of $r$ rows, where $m$ and $r$ are integers. A particular band has less than 100 band members. The director arranges them in a rectangular formation and finds that he has two members left over. If he incre... |
[
{
"from": "human",
"value": "Solve for $x$: $2x + 5 = 15$."
},
{
"from": "gpt",
"value": "To solve the equation $2x + 5 = 15$, we subtract 5 from both sides to get $2x = 10$, then divide by 2 to find $x = 5$."
}
] |
[
{
"from": "human",
"value": "Find the derivative of $f(x) = 3x^2 + 2x - 4$."
},
{
"from": "gpt",
"value": "The derivative of $f(x) = 3x^2 + 2x - 4$ is $f'(x) = 6x + 2$."
}
] |
[
{
"from": "human",
"value": "Evaluate the integral $\\int_0^1 2x \\, dx$."
},
{
"from": "gpt",
"value": "The integral of $2x$ from 0 to 1 is $\\left[ x^2 \\right]_0^1 = 1^2 - 0^2 = 1$."
}
] |
[
{
"from": "human",
"value": "What is the sum of the first 10 positive integers?"
},
{
"from": "gpt",
"value": "The sum of the first 10 positive integers is $\\frac{10(10 + 1)}{2} = 55$."
}
] |
[
{
"from": "human",
"value": "Solve the quadratic equation $x^2 - 5x + 6 = 0$."
},
{
"from": "gpt",
"value": "The solutions of $x^2 - 5x + 6 = 0$ are $x = 2$ and $x = 3$."
}
] |
[
{
"from": "human",
"value": "What is the value of $\\sin(\\frac{\\pi}{6})$?"
},
{
"from": "gpt",
"value": "$\\sin(\\frac{\\pi}{6}) = \\frac{1}{2}$."
}
] |
[
{
"from": "human",
"value": "Find the limit: $\\lim_{x \\to 0} \\frac{\\sin x}{x}$."
},
{
"from": "gpt",
"value": "The limit is $\\lim_{x \\to 0} \\frac{\\sin x}{x} = 1$."
}
] |
[
{
"from": "human",
"value": "What is the area of a circle with radius $r$?"
},
{
"from": "gpt",
"value": "The area of a circle with radius $r$ is $\\pi r^2$."
}
] |
[
{
"from": "human",
"value": "Calculate the derivative of $f(x) = e^x$."
},
{
"from": "gpt",
"value": "The derivative of $f(x) = e^x$ is $f'(x) = e^x$."
}
] |
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