Test your basic knowledge |

AP Calculus Formulas

Instructions:

Answer 50 questions in 15 minutes.
If you are not ready to take this test, you can study here.
Match each statement with the correct term.
Don't refresh. All questions and answers are randomly picked and ordered every time you load a test.

This is a study tool. The 3 wrong answers for each question are randomly chosen from answers to other questions. So, you might find at times the answers obvious, but you will see it re-enforces your understanding as you take the test each time.

1. cos²x + sin²x
1/((ln a) x)
If two functions - f and g - are differentiable - then d/dx[ f(x) / g(x) ] = [g(x)f'(x) - f(x) g'(x)] / [g(x)]²
1
1. Given - Want - Sketch 2. Write an equation using variables given/to be determined 3. Differentiate w.r.t. time (using chain rule) 4. Plug in & solve

2. d/dx[ln x]
1/x - x>0
e^x
-sin x
-1

3. 1 + cot²x
csc²x
x values where f'(x) is zero or undefined.
u' (ln a) a^u
ln m - ln n

4. Average speed
Derivative of Position - s'(t)
?s/?t
0
1

5. d/dx[cos x]
Differentiability implies continuity - but continuity does not necessarily imply differentiability.
-sin x
f'(x) = lim as x ? c of [ f(x) - f(c) ] / [ x - c]
S = 4 pi r^2

6. sin²x
-csc x cot x
(1 - cos 2x) / 2
?s/?t
f'(x) = lim as ?x ? 0 of [ f(x + ?x) - f(x) ] / ?x

7. d/dx[arccot x]
1/x - x>0
1 / tan x = cos x / sin x
n ln m
-1/(1+x²)

8. cos p
1. f(x) is continuous on the closed interval (a -b) 2. The limit from the right as x approaches a of f(x) is f(a) 3. The limit from the left as x approaches b of f(x) is f(b)
u'/u - u > 0
0
-1

9. d/dx[arccos x]
-1/v(1-x²)
1
1/2
f is an odd function

10. d/dx[cot x]
sin x / cos x
-csc² x
If f(x) is continuous on a closed interval [a -b] and k is any number between f(a) and f(b) - then there is at least one number c in [a -b] such that f(c) = k.
1. f(x) is continuous on the closed interval (a -b) 2. The limit from the right as x approaches a of f(x) is f(a) 3. The limit from the left as x approaches b of f(x) is f(b)

11. Volume of a Sphere
-csc² x
1/v(1-x²)
1/2
V = 4/3 pi r^3

12. Volume of a right circular cylinder
0
pr²h
0/0
1 / tan x = cos x / sin x

13. Sum and Difference Rules for Derivatives

14. d/dx[x]
-sin x
1
s(t) = -16t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
v3s² / 4

15. ln 1
cos²x - sin²x
-1
v2/2
0

16. Intermediate Value Theorem
1/(1+x²)
ln m + ln n
-sin x
If f(x) is continuous on a closed interval [a -b] and k is any number between f(a) and f(b) - then there is at least one number c in [a -b] such that f(c) = k.

17. Continuity & differentiability
If two functions - f and g - are differentiable - then d/dx[ f(x) / g(x) ] = [g(x)f'(x) - f(x) g'(x)] / [g(x)]²
Differentiability implies continuity - but continuity does not necessarily imply differentiability.
v3/2
0

18. d/dx[tan x]
sec² x
-csc x cot x
1 / tan x = cos x / sin x
1/x - x>0

19. d/dx[ f(x) g(x) ]

20. Critical number

21. Circumference of a circle
f'(g(x))g'(x)
2pr
pr²
-csc² x

22. cos p/6
If f(x) is continuous on a closed interval [a -b] and k is any number between f(a) and f(b) - then there is at least one number c in [a -b] such that f(c) = k.
0
cos x
v3/2

23. sin 3p/2
-1
g'(x) = 1/f'(g(x)) - f'(g(x)) cannot = 0
0/0
f(x) g'(x) + g(x) f'(x)

24. d/dx[ f(x) / g(x) ]

25. d/dx[arctan x]
s(t) = -16t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
If f is continuous on the closed interval [a -b] then it must have both a minimum and maximum on [a -b].
0
1/(1+x²)

26. sin(2x)
1
1/x - x>0
u'/u - u > 0
2 sin x cos x

27. d/dx[e^x]
e^x
d/dx[c] = 0
-1/(1+x²)
v3s² / 4

28. d/dx[log_a x]
v2/2
1 / sin x
1/((ln a) x)
f(x) is continuous if for every point on the interval (a -b) the conditions for continuity at a point are satisfied.

29. Guidelines for solving related rates problems
d/dx[x^n]=nx^(n-1)
If f is continuous on the closed interval [a -b] then it must have both a minimum and maximum on [a -b].
1. Given - Want - Sketch 2. Write an equation using variables given/to be determined 3. Differentiate w.r.t. time (using chain rule) 4. Plug in & solve
-1/(|x|v(x²-1))

30. Extreme Value Theorem
s(t) = -16t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
If f is continuous on the closed interval [a -b] then it must have both a minimum and maximum on [a -b].
1. Differentiate both sides w.r.t. x 2. Move all y' terms to one side & other terms to the other 3. Factor out y' 4. Divide to solve for y'
d/dx[c] = 0

31. Position function of a falling object (with acceleration in m/s²)
1 / tan x = cos x / sin x
x values where f'(x) is zero or undefined.
s(t) = -4.9t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
csc²x

32. cos(2x)
d/dx[f(x) ± g(x)] = f'(x) ± g'(x)
1/2
cos²x - sin²x
Slope of a function at a point/slope of the tangent line to a function at a point

33. d/dx[a^u]

34. The Quotient Rule

35. Rolle's Theorem

36. Limit Definition of a Derivative

37. sin p/2
1
u'/((ln a) u)
V = 4/3 pi r^3
-1

38. d/dx[arcsin x]
1/v(1-x²)
1
1
s(t) = -16t²+ v0t + s0 - v0 = initial velocity - s0 = initial height

39. sin p/6
1
1/(1+x²)
pr²h
1/2

40. d/dx[log_a u]

41. ln (m/n)
d/dx[cf(x)] = c f'(x)
0
ln m - ln n
1/v(1-x²)

42. sin p/4
v2/2
2 sin x cos x
sec² x
?s/?t

43. d/dx[a^x]
1
s(t) = -16t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
(ln a) a^x
f(x) is continuous if for every point on the interval (a -b) the conditions for continuity at a point are satisfied.

44. tan x
sec² x
1/2
1
sin x / cos x

45. d/dx[sin x]
If two functions - f and g - are differentiable - then d/dx[ f(x) g(x) ] = f(x) g'(x) + g(x) f'(x)
cos x
-csc x cot x
n ln m

46. The limit as x approaches 0 of sin x / x
-sin x
If two functions - f and g - are differentiable - then d/dx[ f(x) g(x) ] = f(x) g'(x) + g(x) f'(x)
Derivative of position at a point
1

47. Continuity on a closed interval - [a -b]
x values where f'(x) is zero or undefined.
0/0
1. f(x) is continuous on the closed interval (a -b) 2. The limit from the right as x approaches a of f(x) is f(a) 3. The limit from the left as x approaches b of f(x) is f(b)
0

48. cos p/4
(1 + cos 2x) / 2
-1/v(1-x²)
v2/2
1/x - x>0

49. Continuity at a point (x = c)
If f(x) is continuous on a closed interval [a -b] and k is any number between f(a) and f(b) - then there is at least one number c in [a -b] such that f(c) = k.
1. f(x) is defined at f(c) 2. The limit as x approaches c of f(x) exists 3. The limit as x approaches c of f(x) = f(c)
f(x) g'(x) + g(x) f'(x)
1 / tan x = cos x / sin x

50. Instantaneous velocity
Derivative of position at a point
1. f(x) is defined at f(c) 2. The limit as x approaches c of f(x) exists 3. The limit as x approaches c of f(x) = f(c)
s(t) = -16t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
1. Differentiate both sides w.r.t. x 2. Move all y' terms to one side & other terms to the other 3. Factor out y' 4. Divide to solve for y'