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. Guidelines for implicit differentiation

2. sin p
0
-1/v(1-x²)
v3s² / 4
Limits

3. cos p/6
v3/2
s(t) = -4.9t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
u'/u - u > 0
x values where f'(x) is zero or undefined.

4. d/dx[arctan x]
u' e^u
ln m - ln n
1/(1+x²)
[g(x)f'(x) - f(x) g'(x)] / [g(x)]²

5. Intermediate Value Theorem
f'(g(x))g'(x)
1 / sin x
1 / cos 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.

6. Position function of a falling object (with acceleration in m/s²)
Derivative of position at a point
-csc² x
s(t) = -4.9t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
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).

7. Extreme Value Theorem
If f is continuous on the closed interval [a -b] then it must have both a minimum and maximum on [a -b].
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/(|x|v(x²-1))
1 / cos x

8. d/dx[x]
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
Differentiability implies continuity - but continuity does not necessarily imply differentiability.
1/v(1-x²)

9. Guidelines for solving related rates problems
2 sin x cos x
f'(x) = lim as x ? c of [ f(x) - f(c) ] / [ x - c]
g'(x) = 1/f'(g(x)) - f'(g(x)) cannot = 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

10. The limit as x approaches 0 of sin x / x
1
V = 4/3 pi r^3
Slope of a function at a point/slope of the tangent line to a function at a point
-1/v(1-x²)

11. 1 + tan²x
sec²x
-1/v(1-x²)
1
sin x / cos x

12. d/dx[ln u]

13. Volume of a Sphere
V = 4/3 pi r^3
1/v(1-x²)
sec x tan x
1

14. sin p/4
d/dx[c] = 0
v2/2
V = 4/3 pi r^3
-1/(1+x²)

15. d/dx[a^x]
x values where f'(x) is zero or undefined.
v3s² / 4
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
(ln a) a^x

16. d/dx[cos x]
-sin x
1
-csc² x
(1 + cos 2x) / 2

17. ln 1
u'/((ln a) u)
0
u'/u - u > 0
1 / tan x = cos x / sin x

18. sin p/3
v3/2
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

19. d/dx[arcsin x]
1/v(1-x²)
f is an even function
sin x / cos x
Differentiability implies continuity - but continuity does not necessarily imply differentiability.

20. Constant Multiple Rule for Derivatives

21. d/dx[a^u]

22. ln (mn)
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
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)
ln m + ln n

23. Critical number

24. sin(2x)
1
(1 + cos 2x) / 2
2 sin x cos x
u'/((ln a) u)

25. sec x
1 / cos x
d/dx[c] = 0
f is an odd function
0

26. d/dx[arccsc x]
1/2
-1/(|x|v(x²-1))
1
d/dx[cf(x)] = c f'(x)

27. d/dx[tan x]
2pr
ln m + ln n
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)
sec² x

28. d/dx[csc x]
-1/v(1-x²)
sec x tan x
d/dx[x^n]=nx^(n-1)
-csc x cot x

29. cos p/4
-1/(|x|v(x²-1))
v2/2
2pr
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)

30. The limit as x approaches 0 of (1 - cos x) / x
v3/2
cos²x - sin²x
0
1/((ln a) x)

31. d/dx[arcsec x]
If two functions - f and g - are differentiable - then d/dx[ f(x) g(x) ] = f(x) g'(x) + g(x) f'(x)
V = 4/3 pi r^3
0
1/(|x|v(x²-1))

32. d/dx[e^u]

33. Surface Area of a Sphere
pr²
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.
S = 4 pi r^2
v3s² / 4

34. Circumference of a circle
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)
2pr
s(t) = -16t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
1

35. Position function of a falling object (with acceleration in ft/s²)
If f is continuous on the closed interval [a -b] then it must have both a minimum and maximum on [a -b].
f is an even function
s(t) = -16t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
sec²x

36. Limit Definition of a Derivative

37. Rolle's Theorem

38. cos(2x)
cos²x - sin²x
Derivative of position at a point
sec²x
f(x) g'(x) + g(x) f'(x)

39. sin p/2
pr²h/3
-csc x cot x
0
1

40. Chain Rule: d/dx[f(g(x))] =

41. The Product Rule

42. Volume of a right circular cylinder
v3/2
-csc x cot x
pr²h
Differentiability implies continuity - but continuity does not necessarily imply differentiability.

43. cos p
-1
u'/u - u > 0
-1/(1+x²)
v3s² / 4

44. Continuity at a point (x = c)
1
If two functions - f and g - are differentiable - then d/dx[ f(x) g(x) ] = f(x) g'(x) + g(x) f'(x)
-csc² 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)

45. Continuity on an open interval - (a -b)
If two functions - f and g - are differentiable - then d/dx[ f(x) g(x) ] = f(x) g'(x) + g(x) f'(x)
pr²h
f(x) is continuous if for every point on the interval (a -b) the conditions for continuity at a point are satisfied.
cos²x - sin²x

46. ln (m/n)
S = 4 pi r^2
1/2
ln m - ln n
g'(x) = 1/f'(g(x)) - f'(g(x)) cannot = 0

47. cos p/2
n ln m
s(t) = -16t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
ln m - ln n
0

48. ln mn
1
n ln m
-1/v(1-x²)
0

49. d/dx[log_a x]
s(t) = -4.9t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
1/((ln a) x)
1
-1/(|x|v(x²-1))

50. cos 3p/2
cos x
0
Limits
v2/2