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AP Calculus Formulas

Instructions:

Answer 50 questions in 15 minutes.
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Match each statement with the correct term.
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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. Chain Rule: d/dx[f(g(x))] =

2. Velocity - v(t)

3. The Quotient Rule

4. If f(-x) = -f(x)
f is an odd function
pr²h
csc²x
0

5. ln (mn)
ln m + ln n
0
If f is continuous on the closed interval [a -b] then it must have both a minimum and maximum on [a -b].
2pr

6. sin 0
x values where f'(x) is zero or undefined.
f is an even function
0
1

7. cos p/2
1 / sin x
0
csc²x
1

8. Volume of a Sphere
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
V = 4/3 pi r^3
[g(x)f'(x) - f(x) g'(x)] / [g(x)]²
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)

9. Area of an equilateral triangle
v3/2
f is an even function
(ln a) a^x
v3s² / 4

10. If f(-x) = f(x)
sin x / cos x
-1
0
f is an even function

11. d/dx[sin x]
cos x
[g(x)f'(x) - f(x) g'(x)] / [g(x)]²
v3/2
1/2

12. sin p
-csc² x
1/x - x>0
0
1

13. 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)
f is an even function
cos x
d/dx[f(x) ± g(x)] = f'(x) ± g'(x)

14. sec x
sec² x
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'
csc²x
1 / cos x

15. Circumference of a circle
2pr
1/2
cos²x - sin²x
1

16. cot x
1 / tan x = cos x / sin x
pr²
0
f(x) is continuous if for every point on the interval (a -b) the conditions for continuity at a point are satisfied.

17. Average speed
-sin x
?s/?t
-1/v(1-x²)
s(t) = -4.9t²+ v0t + s0 - v0 = initial velocity - s0 = initial height

18. Continuity & differentiability
(ln a) a^x
d/dx[x^n]=nx^(n-1)
g'(x) = 1/f'(g(x)) - f'(g(x)) cannot = 0
Differentiability implies continuity - but continuity does not necessarily imply differentiability.

19. Instantaneous velocity
Derivative of position at a point
0
[g(x)f'(x) - f(x) g'(x)] / [g(x)]²
-1

20. Alternate Limit Definition of a derivative

21. Derivative
u'/((ln a) u)
cos²x - sin²x
Slope of a function at a point/slope of the tangent line to a function at a point
s(t) = -16t²+ v0t + s0 - v0 = initial velocity - s0 = initial height

22. 1 + cot²x
?s/?t
0
csc²x
0

23. Rolle's Theorem

24. cos 3p/2
s(t) = -16t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
-sin x
pr²h/3
0

25. d/dx[cos x]
-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)
0
1 / tan x = cos x / sin x

26. cos p
0
-1
-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)

27. Area of a circle
pr²
1/2
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

28. d/dx[arccot x]
u'/u - u > 0
-1/(1+x²)
0
v3/2

29. The limit as x approaches 0 of sin x / x
1 / sin x
1
1/2
u' (ln a) a^u

30. cos p/6
s(t) = -16t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
1/(1+x²)
v3/2
csc²x

31. cos p/3
1/2
-1
d/dx[x^n]=nx^(n-1)
f'(g(x))g'(x)

32. Volume of a cone
1/2
pr²h/3
v3/2
Slope of a function at a point/slope of the tangent line to a function at a point

33. d/dx[arccos x]
f'(x) = lim as ?x ? 0 of [ f(x + ?x) - f(x) ] / ?x
-1
(ln a) a^x
-1/v(1-x²)

34. d/dx[cot x]
v3/2
2 sin x cos x
-csc² x
1

35. cos 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)
u'/u - u > 0
1
1 / tan x = cos x / sin x

36. d/dx[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).
(ln a) a^x
u' (ln a) a^u
(1 - cos 2x) / 2

37. sin²x
0/0
0
pr²h
(1 - cos 2x) / 2

38. d/dx[x]
0
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'
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

39. Power Rule for Derivatives
Derivative of Position - s'(t)
1 / tan x = cos x / sin x
d/dx[x^n]=nx^(n-1)
1

40. Sum and Difference Rules for Derivatives

41. d/dx[arcsec x]
1/(|x|v(x²-1))
g'(x) = 1/f'(g(x)) - f'(g(x)) cannot = 0
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

42. d/dx[arccsc x]
0
u' (ln a) a^u
-1/(|x|v(x²-1))
x values where f'(x) is zero or undefined.

43. Position function of a falling object (with acceleration in ft/s²)
f'(g(x))g'(x)
s(t) = -16t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
x values where f'(x) is zero or undefined.
d/dx[x^n]=nx^(n-1)

44. Intermediate Value Theorem
-1
v3/2
f is an even function
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.

45. Guidelines for implicit differentiation

46. ln (m/n)
ln m - ln n
n ln m
1
u'/u - u > 0

47. ln mn
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.
n ln m
pr²h
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

48. cos p/4
-1/(|x|v(x²-1))
If f is continuous on the closed interval [a -b] then it must have both a minimum and maximum on [a -b].
-1/v(1-x²)
v2/2

49. ln 1
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)
e^x
0
f is an odd function

50. d/dx[csc x]
sin x / cos x
sec²x
-csc x cot x
x values where f'(x) is zero or undefined.