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. The Product Rule

2. sin 0
pr²
0
-sin x
e^x

3. cos p/4
e^x
v3/2
v2/2
1/(|x|v(x²-1))

4. Instantaneous velocity
1 / cos x
u' e^u
S = 4 pi r^2
Derivative of position at a point

5. d/dx[arccsc x]
1/2
-1/(|x|v(x²-1))
1/x - x>0
1/(|x|v(x²-1))

6. Area of an equilateral triangle
1/2
v3s² / 4
sec x tan x
1

7. d/dx[ln u]

8. Intermediate Value Theorem
(ln a) a^x
pr²h
Slope of a function at a point/slope of the tangent line to a function at a point
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.

9. cos p/6
v3/2
f is an even function
1/2
g'(x) = 1/f'(g(x)) - f'(g(x)) cannot = 0

10. d/dx[a^x]
(ln a) a^x
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).
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
-csc x cot x

11. d/dx[arctan x]
Derivative of Position - s'(t)
sec²x
g'(x) = 1/f'(g(x)) - f'(g(x)) cannot = 0
1/(1+x²)

12. Continuity on an open interval - (a -b)
V = 4/3 pi r^3
[g(x)f'(x) - f(x) g'(x)] / [g(x)]²
f(x) is continuous if for every point on the interval (a -b) the conditions for continuity at a point are satisfied.
ln m - ln n

13. ln (m/n)
cos²x - sin²x
1/(1+x²)
(1 + cos 2x) / 2
ln m - ln n

14. cos 0
1
f(x) g'(x) + g(x) f'(x)
-1/v(1-x²)
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).

15. Guidelines for implicit differentiation

16. Volume of a right circular cylinder
Derivative of Position - s'(t)
pr²h
V = 4/3 pi r^3
v3s² / 4

17. d/dx[cot x]
f'(x) = lim as x ? c of [ f(x) - f(c) ] / [ x - c]
-csc² x
Differentiability implies continuity - but continuity does not necessarily imply differentiability.
d/dx[cf(x)] = c f'(x)

18. Continuity & differentiability
u' (ln a) a^u
Differentiability implies continuity - but continuity does not necessarily imply differentiability.
pr²h
if f(x) is continuous and differentiable - slope of tangent line equals slope of secant line at least once in the interval (a - b) - f '(c) = [f(b) - f(a)]/(b - a)

19. Sum and Difference Rules for Derivatives

20. sin(2x)
s(t) = -16t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
ln m - ln n
1 / cos x
2 sin x cos x

21. d/dx[sec x]
pr²h/3
f is an odd function
sec x tan x
1 / tan x = cos x / sin x

22. 1 + cot²x
Differentiability implies continuity - but continuity does not necessarily imply differentiability.
cos x
csc²x
1

23. Derivative of a constant
d/dx[c] = 0
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
v3/2
Differentiability implies continuity - but continuity does not necessarily imply differentiability.

24. Rolle's Theorem

25. Position function of a falling object (with acceleration in ft/s²)
s(t) = -16t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
ln m - ln n
1/2
1/v(1-x²)

26. sin p/3
1/(1+x²)
d/dx[cf(x)] = c f'(x)
v3/2
-1

27. d/dx[e^x]
sec² x
e^x
If two functions - f and g - are differentiable - then d/dx[ f(x) g(x) ] = f(x) g'(x) + g(x) f'(x)
-1/v(1-x²)

28. Continuity on a closed interval - [a -b]
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)
f is an even function
2 sin x cos x
-csc x cot x

29. d/dx[a^u]

30. d/dx[arcsin x]
1/v(1-x²)
-1
?s/?t
1/(1+x²)

31. The Quotient Rule

32. If f(-x) = f(x)
ln m + ln n
if f(x) is continuous and differentiable - slope of tangent line equals slope of secant line at least once in the interval (a - b) - f '(c) = [f(b) - f(a)]/(b - a)
f(x) g'(x) + g(x) f'(x)
f is an even function

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

34. cos p/2
0
1
cos x
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)

35. cos(2x)
ln m - ln n
cos²x - sin²x
1
pr²h

36. Derivative
d/dx[x^n]=nx^(n-1)
1
d/dx[f(x) ± g(x)] = f'(x) ± g'(x)
Slope of a function at a point/slope of the tangent line to a function at a point

37. d/dx[cos x]
(1 + cos 2x) / 2
ln m + ln n
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'
-sin x

38. d/dx[x]
1/x - x>0
sec x tan x
V = 4/3 pi r^3
1

39. cos p
-1
csc²x
1
[g(x)f'(x) - f(x) g'(x)] / [g(x)]²

40. Continuity at a point (x = c)
ln m + ln n
sec x tan x
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)
1

41. tan x
f(x) is continuous if for every point on the interval (a -b) the conditions for continuity at a point are satisfied.
(1 - cos 2x) / 2
sin x / cos x
cos x

42. Power Rule for Derivatives
-1
d/dx[x^n]=nx^(n-1)
Derivative of Position - s'(t)
-1/(1+x²)

43. sin p/6
n ln m
v2/2
u' e^u
1/2

44. sec x
1 / tan x = cos x / sin x
1 / cos x
u' e^u
Differentiability implies continuity - but continuity does not necessarily imply differentiability.

45. Extreme Value Theorem
1 / sin x
If f is continuous on the closed interval [a -b] then it must have both a minimum and maximum on [a -b].
0
if f(x) is continuous and differentiable - slope of tangent line equals slope of secant line at least once in the interval (a - b) - f '(c) = [f(b) - f(a)]/(b - a)

46. Constant Multiple Rule for Derivatives

47. Surface Area of a Sphere
S = 4 pi r^2
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
Derivative of Position - s'(t)
0/0

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

49. The limit as x approaches 0 of sin x / x
Limits
1/v(1-x²)
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

50. d/dx[arcsec x]
d/dx[x^n]=nx^(n-1)
1/(|x|v(x²-1))
1
1 / cos x