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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. d/dx[a^x]
(ln a) a^x
-csc x cot 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'
[g(x)f'(x) - f(x) g'(x)] / [g(x)]²

2. cos p
-1
-csc² x
(1 + cos 2x) / 2
1/v(1-x²)

3. The Product Rule

4. Velocity - v(t)

5. d/dx[arccot x]
1/x - x>0
-1/(1+x²)
2 sin x cos x
1 / tan x = cos x / sin x

6. If f(-x) = f(x)
Derivative of position at a point
d/dx[c] = 0
f is an even function
f(x) is continuous if for every point on the interval (a -b) the conditions for continuity at a point are satisfied.

7. ln 1
-1/v(1-x²)
1
1/(|x|v(x²-1))
0

8. sin p/2
Derivative of Position - s'(t)
v2/2
ln m - ln n
1

9. Rolle's Theorem

10. cot x
1/((ln a) x)
v3/2
[g(x)f'(x) - f(x) g'(x)] / [g(x)]²
1 / tan x = cos x / sin x

11. Volume of a Sphere
V = 4/3 pi r^3
1/(1+x²)
1 / tan x = cos x / sin 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).

12. Mean Value Theorem

13. tan x
pr²
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
pr²h
sin x / cos x

14. The Quotient Rule

15. Continuity on a closed interval - [a -b]
Slope of a function at a point/slope of the tangent line to a function at a point
g'(x) = 1/f'(g(x)) - f'(g(x)) cannot = 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)
Differentiability implies continuity - but continuity does not necessarily imply differentiability.

16. d/dx[a^u]

17. Guidelines for solving related rates problems
-sin x
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
u'/((ln a) u)

18. d/dx[e^u]

19. Surface Area of a Sphere
S = 4 pi r^2
1/2
d/dx[x^n]=nx^(n-1)
0/0

20. cos p/6
v3s² / 4
0
v3/2
e^x

21. sec x
csc²x
x values where f'(x) is zero or undefined.
1 / cos x
0

22. sin p
u'/((ln a) u)
0
1/x - x>0
1 / sin x

23. d/dx[cot x]
[g(x)f'(x) - f(x) g'(x)] / [g(x)]²
0/0
-csc² x
1/(|x|v(x²-1))

24. Continuity on an open interval - (a -b)
1 / sin x
?s/?t
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

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

26. 1 + tan²x
0/0
csc²x
-csc x cot x
sec²x

27. cos²x + sin²x
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
If two functions - f and g - are differentiable - then d/dx[ f(x) g(x) ] = f(x) g'(x) + g(x) f'(x)
1
0

28. sin(2x)
Limits
0
2 sin x cos x
Slope of a function at a point/slope of the tangent line to a function at a point

29. sin 0
0
-1
pr²h
2pr

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

31. sin 3p/2
-1
f is an even function
u'/((ln a) u)
1/2

32. If f(-x) = -f(x)
s(t) = -4.9t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
f is an odd function
1 / sin x
-csc x cot x

33. Constant Multiple Rule for Derivatives

34. d/dx[ln x]
1/x - x>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)
Derivative of position at a point

35. sin²x
u' (ln a) a^u
1/((ln a) x)
e^x
(1 - cos 2x) / 2

36. d/dx[sin x]
d/dx[f(x) ± g(x)] = f'(x) ± g'(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/(|x|v(x²-1))
cos x

37. d/dx[csc x]
(1 - cos 2x) / 2
-csc x cot x
0
sec²x

38. Area of an equilateral triangle
f is an even function
1/2
-csc x cot x
v3s² / 4

39. Position function of a falling object (with acceleration in ft/s²)
d/dx[f(x) ± g(x)] = f'(x) ± g'(x)
v2/2
s(t) = -16t²+ v0t + s0 - v0 = initial velocity - s0 = initial height
v3/2

40. ln (mn)
-1
d/dx[f(x) ± g(x)] = f'(x) ± g'(x)
-1/(1+x²)
ln m + ln n

41. d/dx[arcsin x]
d/dx[f(x) ± g(x)] = f'(x) ± g'(x)
1/v(1-x²)
sec x tan x
Limits

42. Volume of a cone
e^x
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
pr²h/3
f(x) is continuous if for every point on the interval (a -b) the conditions for continuity at a point are satisfied.

43. d/dx[cos x]
-sin x
Differentiability implies continuity - but continuity does not necessarily imply differentiability.
1 / sin x
sec x tan x

44. Extreme Value Theorem
g'(x) = 1/f'(g(x)) - f'(g(x)) cannot = 0
sec x tan x
If f is continuous on the closed interval [a -b] then it must have both a minimum and maximum on [a -b].
1

45. sin p/3
e^x
u'/((ln a) u)
v3/2
s(t) = -4.9t²+ v0t + s0 - v0 = initial velocity - s0 = initial height

46. Derivative of a constant
1/v(1-x²)
d/dx[c] = 0
-sin x
v3/2

47. Volume of a right circular cylinder
1
ln m + ln n
pr²h
[g(x)f'(x) - f(x) g'(x)] / [g(x)]²

48. Critical number

49. Sum and Difference Rules for Derivatives

50. cos²x
1 / sin x
sec x tan x
(1 + cos 2x) / 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)