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. d/dx[tan x]
v3/2
Differentiability implies continuity - but continuity does not necessarily imply differentiability.
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.
sec x

2. Rolle's Theorem
v2/2
cosx - sinx
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).
Derivative of position at a point

3. cos 0
f is an even function
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)
1
1 / cos x

4. Position function of a falling object (with acceleration in m/s)
Differentiability implies continuity - but continuity does not necessarily imply differentiability.
sin x / cos x
s(t) = -4.9t+ v0t + s0 - v0 = initial velocity - s0 = initial height
n ln m

5. Intermediate Value Theorem
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.
Limits
1
1/((ln a) x)

6. cos p
0
cosx - sinx
-1
u'/((ln a) u)

7. Volume of a cone
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'
ln m - ln n
prh/3
(1 + cos 2x) / 2

8. ln (m/n)
-1
0
2 sin x cos x
ln m - ln n

9. The Quotient Rule
1/(|x|v(x-1))
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)]
0
v2/2

10. cos p/2
0
(ln a) a^x
ln m - ln n
[g(x)f'(x) - f(x) g'(x)] / [g(x)]

11. Constant Multiple Rule for Derivatives
1/v(1-x)
S = 4 pi r^2
d/dx[cf(x)] = c f'(x)
f is an odd function

12. Chain Rule: d/dx[f(g(x))] =
1/2
[g(x)f'(x) - f(x) g'(x)] / [g(x)]
0
f'(g(x))g'(x)

13. d/dx[sin 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)
cos x
f(x) is continuous if for every point on the interval (a -b) the conditions for continuity at a point are satisfied.
f'(x) = lim as x ? c of [ f(x) - f(c) ] / [ x - c]

14. d/dx[a^x]
1/((ln a) x)
1
(1 + cos 2x) / 2
(ln a) a^x

15. sin p/6
-csc x
f is an even function
-1
1/2

16. sin p
(1 - cos 2x) / 2
0
1/(1+x)
cos x

17. Position function of a falling object (with acceleration in ft/s)
s(t) = -16t+ v0t + s0 - v0 = initial velocity - s0 = initial height
sec x tan x
-1/(|x|v(x-1))
?s/?t

18. Critical number
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'
-1
1/2
x values where f'(x) is zero or undefined.

19. d/dx[arccot x]
pr
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'
cosx - sinx
-1/(1+x)

20. d/dx[cot x]
f'(x) = lim as ?x ? 0 of [ f(x + ?x) - f(x) ] / ?x
-csc x
?s/?t
u'/u - u > 0

21. Continuity at a point (x = c)
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)]
Slope of a function at a point/slope of the tangent line to a function at a point
-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)

22. cosx
f(x) g'(x) + g(x) f'(x)
sec x 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

23. d/dx[cos x]
ln m - ln n
-sin x
1
0

24. Power Rule for Derivatives
d/dx[x^n]=nx^(n-1)
1/2
u'/u - u > 0
1

25. Derivative of a constant
d/dx[c] = 0
Slope of a function at a point/slope of the tangent line to a function 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)
f'(x) = lim as x ? c of [ f(x) - f(c) ] / [ x - c]

26. Instantaneous velocity
Derivative of position at a point
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)
Slope of a function at a point/slope of the tangent line to a function at a point
2 sin x cos x

27. d/dx[ln u]
(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.
Derivative of Position - s'(t)
u'/u - u > 0

28. Alternate Limit Definition of a derivative
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'
2 sin x cos x
f'(x) = lim as x ? c of [ f(x) - f(c) ] / [ x - c]
prh/3

29. d/dx[e^x]
e^x
0
-csc x
If f is continuous on the closed interval [a -b] then it must have both a minimum and maximum on [a -b].

30. Volume of a right circular cylinder
Derivative of position at a point
sin x / cos x
-1/(|x|v(x-1))
prh

31. Area of a circle
u' e^u
1 / sin x
pr
f is an even function

32. d/dx[e^u]
u' e^u
f'(x) = lim as x ? c of [ f(x) - f(c) ] / [ x - c]
0
0

33. Mean Value Theorem
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)
Derivative of Position - s'(t)
If two functions - f and g - are differentiable - then d/dx[ f(x) g(x) ] = f(x) g'(x) + g(x) f'(x)
(ln a) a^x

34. csc x
1 / sin x
u' e^u
u'/((ln a) u)
f(x) is continuous if for every point on the interval (a -b) the conditions for continuity at a point are satisfied.

35. Area of an equilateral triangle
1/x - x>0
s(t) = -16t+ v0t + s0 - v0 = initial velocity - s0 = initial height
sec x tan x
v3s / 4

36. Guidelines for implicit differentiation
f is an odd 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'
sec x tan x
secx

37. d/dx[x]
If f is continuous on the closed interval [a -b] then it must have both a minimum and maximum on [a -b].
1
Slope of a function at a point/slope of the tangent line to a function at a point
0

38. sec x
0
1 / cos x
cosx - sinx
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)]

39. cosx + sinx
0/0
1/x - x>0
1
0

40. cos 3p/2
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)
prh
0
-csc x

41. cos(2x)
1
1 / tan x = cos x / sin x
cosx - sinx
1/2

42. Indeterminate form
v3/2
(ln a) a^x
0/0
If two functions - f and g - are differentiable - then d/dx[ f(x) g(x) ] = f(x) g'(x) + g(x) f'(x)

43. Continuity & differentiability
sin x / cos x
1/2
?s/?t
Differentiability implies continuity - but continuity does not necessarily imply differentiability.

44. How to get from precalculus to calculus
Limits
(ln a) a^x
1
Derivative of Position - s'(t)

45. d/dx[a^u]
u' (ln a) a^u
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)
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.
prh/3

46. d/dx[log_a u]
(1 + cos 2x) / 2
u'/((ln a) u)
f'(g(x))g'(x)
e^x

47. d/dx[arccsc x]
-1/(|x|v(x-1))
v3s / 4
1
f'(x) = lim as x ? c of [ f(x) - f(c) ] / [ x - c]

48. ln 1
-1/(1+x)
v2/2
v3/2
0

49. Circumference of a circle
ln m + ln n
2pr
1/x - x>0
2 sin x cos x

50. Derivative of an inverse (if g(x) is the inverse of f(x))
ln m + ln n
u' (ln a) a^u
f'(g(x))g'(x)
g'(x) = 1/f'(g(x)) - f'(g(x)) cannot = 0

Test your basic knowledge |

AP Calculus Formulas

Communication skills, Self help skills, Self improvement skills, Productivity skills, Writing skills, Business skills, Thinking skills & More...

502 Pages | Only $12 | PDF / EPub, Kindle Ready

Top Scores

Link to This Test

Related Subjects