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. sin(2x)
ln m - ln n
(1 + cos 2x) / 2
n ln m
2 sin x cos x

2. 1 + cot²x
csc²x
(ln a) a^x
Differentiability implies continuity - but continuity does not necessarily imply differentiability.
-1/(|x|v(x²-1))

3. 1 + tan²x
1/(|x|v(x²-1))
1
csc²x
sec²x

4. cos(2x)
cos²x - sin²x
If two functions - f and g - are differentiable - then d/dx[ f(x) g(x) ] = f(x) g'(x) + g(x) f'(x)
1/2
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'

5. d/dx[ln x]
2pr
x values where f'(x) is zero or undefined.
1/x - x>0
Let f be continuous on [a -b] and differentiable on (a -b) and if f(a)=f(b) then there is at least one number c on (a -b) such that f'(c)=0 (If the slope of the secant is 0 - the derivative must = 0 somewhere in the interval).

6. Sum and Difference Rules for Derivatives

7. csc x
[g(x)f'(x) - f(x) g'(x)] / [g(x)]²
1 / sin x
V = 4/3 pi r^3
f(x) is continuous if for every point on the interval (a -b) the conditions for continuity at a point are satisfied.

8. Guidelines for implicit differentiation

9. Power Rule for Derivatives
v2/2
-sin x
d/dx[x^n]=nx^(n-1)
sec² x

10. Indeterminate form
1/2
0/0
1 / sin x
pr²h

11. d/dx[x]
csc²x
f is an odd function
1
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)

12. Continuity at a point (x = c)
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)
ln m + ln n
1/((ln a) x)
-csc² x

13. sec x
f'(g(x))g'(x)
f is an odd function
S = 4 pi r^2
1 / cos x

14. cot x
0
f(x) is continuous if for every point on the interval (a -b) the conditions for continuity at a point are satisfied.
-csc² x
1 / tan x = cos x / sin x

15. Derivative of an inverse (if g(x) is the inverse of f(x))

16. The limit as x approaches 0 of (1 - cos x) / x
0
v2/2
(1 + cos 2x) / 2
s(t) = -16t²+ v0t + s0 - v0 = initial velocity - s0 = initial height

17. d/dx[tan x]
sec² x
v2/2
0/0
(1 - cos 2x) / 2

18. Position function of a falling object (with acceleration in ft/s²)
f is an odd function
s(t) = -16t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
S = 4 pi r^2
d/dx[cf(x)] = c f'(x)

19. The Quotient Rule

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

21. cos p/3
f'(g(x))g'(x)
?s/?t
1/2
u' e^u

22. cos 3p/2
f is an even function
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'
0
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.

23. sin p/2
v3/2
0
-sin x
1

24. cos²x
sec x tan x
0
(1 + cos 2x) / 2
f'(g(x))g'(x)

25. Volume of a Sphere
1/((ln a) x)
0
V = 4/3 pi r^3
v2/2

26. Derivative
f is an odd function
Slope of a function at a point/slope of the tangent line to a function at a point
(ln a) a^x
u' (ln a) a^u

27. The Product Rule

28. d/dx[arccos x]
f'(x) = lim as x ? c of [ f(x) - f(c) ] / [ x - c]
-1/v(1-x²)
cos²x - sin²x
1/(1+x²)

29. Guidelines for solving related rates problems
u'/((ln a) u)
If two functions - f and g - are differentiable - then d/dx[ f(x) g(x) ] = f(x) g'(x) + g(x) f'(x)
ln m - ln n
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

30. Intermediate Value Theorem
1
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.
v3/2
0

31. Continuity & differentiability
0
0
Differentiability implies continuity - but continuity does not necessarily imply differentiability.
?s/?t

32. Critical number

33. cos p/6
0/0
v3/2
-csc x cot x
u'/u - u > 0

34. Circumference of a circle
cos x
s(t) = -4.9t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
f(x) g'(x) + g(x) f'(x)
2pr

35. d/dx[log_a u]

36. ln mn
n ln m
(1 + cos 2x) / 2
0
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.

37. Constant Multiple Rule for Derivatives

38. cos p/2
v3/2
d/dx[cf(x)] = c f'(x)
0
s(t) = -16t²+ v0t + s0 - v0 = initial velocity - s0 = initial height

39. The limit as x approaches 0 of sin x / x
?s/?t
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
u' (ln a) a^u

40. Average speed
?s/?t
pr²h/3
ln m + ln n
u' (ln a) a^u

41. Alternate Limit Definition of a derivative

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

43. Position function of a falling object (with acceleration in m/s²)
ln m + ln n
s(t) = -4.9t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
1 / sin x
(1 + cos 2x) / 2

44. d/dx[a^u]

45. How to get from precalculus to calculus
1
pr²h/3
-1/(1+x²)
Limits

46. tan x
1/((ln a) x)
1
f'(g(x))g'(x)
sin x / cos x

47. d/dx[e^u]

48. ln 1
0
-csc x cot x
-csc² x
1

49. d/dx[sec x]
sec x tan x
n ln m
0
?s/?t

50. ln (mn)
ln m + ln n
f is an even function
u' (ln a) a^u
[g(x)f'(x) - f(x) g'(x)] / [g(x)]²