Math Mix: 62 flashcards
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1. What is the formula for the area of a circle?
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$A = \pi r^2$
2. What is the Pythagorean theorem?
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In a right triangle, $a^2 + b^2 = c^2$, where $c$ is the hypotenuse.
3. What is the quadratic formula?
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$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
4. What is the slope-intercept form of a line?
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$y = mx + b$, where $m$ is slope and $b$ is y-intercept.
5. What is the derivative of $\sin x$?
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$\cos x$
6. What is the derivative of $e^x$?
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$e^x$
7. What is the integral of $\frac{1}{x}$?
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$\ln |x| + C$
8. What is the value of $\sin 30^\circ$?
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$\frac{1}{2}$
9. What is the value of $\cos 60^\circ$?
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$\frac{1}{2}$
10. What is the formula for the volume of a cylinder?
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$V = \pi r^2 h$
11. What is the formula for the surface area of a sphere?
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$A = 4\pi r^2$
12. What is the sum of the interior angles of a triangle?
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$180^\circ$
13. What is the sum of the interior angles of a quadrilateral?
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$360^\circ$
14. What is the formula for the circumference of a circle?
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$C = 2\pi r$
15. What is the formula for the area of a triangle?
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$A = \frac{1}{2}bh$
16. What is the formula for the area of a trapezoid?
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$A = \frac{1}{2}(b_1 + b_2)h$
17. What is the formula for the volume of a rectangular prism?
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$V = lwh$
18. What is the formula for the volume of a sphere?
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$V = \frac{4}{3}\pi r^3$
19. What is the formula for the volume of a cone?
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$V = \frac{1}{3}\pi r^2 h$
20. What is the formula for the volume of a pyramid?
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$V = \frac{1}{3}Bh$
21. What is the formula for the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$?
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$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
22. What is the formula for the midpoint of a segment with endpoints $(x_1, y_1)$ and $(x_2, y_2)$?
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$M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$
23. What is the slope of a line passing through $(x_1, y_1)$ and $(x_2, y_2)$?
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$m = \frac{y_2 - y_1}{x_2 - x_1}$
24. What is the equation of a circle with center $(h, k)$ and radius $r$?
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$(x - h)^2 + (y - k)^2 = r^2$
25. What is the formula for the $n$th term of an arithmetic sequence?
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$a_n = a_1 + (n - 1)d$
26. What is the formula for the sum of the first $n$ terms of an arithmetic series?
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$S_n = \frac{n}{2}(a_1 + a_n)$
27. What is the formula for the $n$th term of a geometric sequence?
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$a_n = a_1 \cdot r^{n-1}$
28. What is the formula for the sum of an infinite geometric series with $|r| < 1$?
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$S = \frac{a_1}{1 - r}$
29. What is the formula for simple interest?
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$I = P r t$
30. What is the formula for compound interest?
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$A = P\left(1 + \frac{r}{n}\right)^{nt}$
31. What is the formula for the probability of an event?
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$P(E) = \frac{\text{number of favorable outcomes}}{\text{total number of outcomes}}$
32. What is the formula for the mean of a data set?
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$\bar{x} = \frac{\sum x_i}{n}$
33. What is the formula for the median of a data set?
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The middle value when data is ordered; if even number of values, average of two middle values.
34. What is the formula for the mode of a data set?
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The value that appears most frequently.
35. What is the formula for the standard deviation of a sample?
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$s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}$
36. What is the formula for the variance of a sample?
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$s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1}$
37. What is the formula for the z-score?
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$z = \frac{x - \mu}{\sigma}$
38. What is the formula for the binomial probability?
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$P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$
39. What is the formula for the combination $\binom{n}{k}$?
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$\binom{n}{k} = \frac{n!}{k!(n-k)!}$
40. What is the formula for the permutation $P(n, k)$?
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$P(n, k) = \frac{n!}{(n-k)!}$
41. What is the formula for the factorial $n!$?
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$n! = n \times (n-1) \times \cdots \times 1$
42. What is the formula for the sum of the first $n$ positive integers?
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$\sum_{i=1}^n i = \frac{n(n+1)}{2}$
43. What is the formula for the sum of the first $n$ squares?
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$\sum_{i=1}^n i^2 = \frac{n(n+1)(2n+1)}{6}$
44. What is the formula for the sum of the first $n$ cubes?
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$\sum_{i=1}^n i^3 = \left( \frac{n(n+1)}{2} \right)^2$
45. What is the formula for the binomial theorem expansion of $(a+b)^n$?
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$(a+b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k$
46. What is the formula for the derivative of $x^n$?
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$\frac{d}{dx} x^n = n x^{n-1}$
47. What is the formula for the integral of $x^n$ (for $n \neq -1$)?
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$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$
48. What is the formula for the derivative of $\cos x$?
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$-\sin x$
49. What is the formula for the derivative of $\tan x$?
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$\sec^2 x$
50. What is the formula for the derivative of $\ln x$?
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$\frac{1}{x}$
51. What is the formula for the product rule?
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$(uv)' = u'v + uv'$
52. What is the formula for the quotient rule?
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$\left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2}$
53. What is the formula for the chain rule?
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$\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)$
54. What is the formula for integration by parts?
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$\int u \, dv = uv - \int v \, du$
55. What is the formula for the area under a curve $y = f(x)$ from $x=a$ to $x=b$?
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$\int_a^b f(x) \, dx$
56. What is the formula for the volume of a solid of revolution using disks?
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$V = \pi \int_a^b [f(x)]^2 \, dx$
57. What is the formula for the arc length of a curve $y = f(x)$ from $x=a$ to $x=b$?
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$L = \int_a^b \sqrt{1 + [f'(x)]^2} \, dx$
58. What is the formula for the surface area of a solid of revolution about the x-axis?
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$S = 2\pi \int_a^b f(x) \sqrt{1 + [f'(x)]^2} \, dx$
59. What is the formula for the slope of a tangent line to $y = f(x)$ at $x = a$?
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$f'(a)$
60. What is the formula for the equation of a tangent line to $y = f(x)$ at $x = a$?
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$y - f(a) = f'(a)(x - a)$
61. What is the formula for the average rate of change of $f$ on $[a, b]$?
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$\frac{f(b) - f(a)}{b - a}$
62. What is the formula for the instantaneous rate of change of $f$ at $x = a$?
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$\lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$
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