AP CALCULUS AB
STUFF YOU MUST
KNOW COLD
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As you prepare your students for the AP Calculus AB Exam, hereโs an adaptation to Sean Birdโs โStuff You Must Know Coldโ handout. (http://covenantchristian.org/bird/Calculus.htm) Formulas, Theorems, and Other concepts necessary for success for the Calculus AB student. You will find additional resources, including a PowerPoint version, a Word document in color, as well as AP Calculus BC topics on his website.
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STUFF YOU MUST KNOW COLD . . .
Basic Derivatives
๐๐๐๐๐๐
(๐๐๐๐) = ๐๐๐๐๐๐โ1
๐๐๐๐๐๐
(sin๐๐) = cos ๐๐
๐๐๐๐๐๐
(cos ๐๐) = โ sin ๐๐
๐๐๐๐๐๐
(tan ๐๐) = sec2 ๐๐
๐๐๐๐๐๐
(cot ๐๐) = โ csc2 ๐๐
๐๐๐๐๐๐
(sec๐๐) = sec ๐๐ tan ๐๐
๐๐๐๐๐๐
(csc๐๐) = โ csc๐๐ cot ๐๐
๐๐๐๐๐๐
(ln๐ข๐ข) =1 ๐ข๐ข
๐๐๐ข๐ข๐๐๐๐
๐๐๐๐๐๐
(๐๐๐ข๐ข) = ๐๐๐ข๐ข ๐๐๐ข๐ข๐๐๐๐
Where u is a function of x, and a is a constant.
Differentiation Rules Chain Rule: ๐๐๐๐๐๐
[๐๐(๐ข๐ข)] = ๐๐โฒ(๐ข๐ข) ๐๐๐ข๐ข๐๐๐๐
OR ๐๐๐๐๐๐๐๐
= ๐๐๐๐๐๐๐ข๐ข
๐๐๐ข๐ข๐๐๐๐
Product Rule: ๐๐๐๐๐๐
(๐ข๐ข๐ข๐ข) = ๐ข๐ข ๐๐๐๐๐๐๐๐
+ ๐ข๐ข ๐๐๐ข๐ข๐๐๐๐
OR ๐ข๐ข ๐ข๐ขโฒ + ๐ข๐ข ๐ข๐ขโฒ Quotient Rule: ๐๐๐๐๐๐๏ฟฝ๐ข๐ข๐๐๏ฟฝ =
๐๐ ๐๐๐๐๐๐๐๐โ ๐ข๐ข๐๐๐๐๐๐๐๐๐๐2
OR ๐๐ ๐ข๐ขโฒโ ๐ข๐ข ๐๐โฒ
๐๐2
Intermediate Value Theorem
If the function ๐๐(๐๐) is continuous on [๐๐, ๐๐], and y is a number between ๐๐(๐๐) and ๐๐(๐๐), then there exists at least one number ๐๐ = ๐๐ in the open interval (๐๐, ๐๐) such that ๐๐(๐๐) = ๐ฆ๐ฆ.
Mean Value Theorem
If the function ๐๐(๐๐) is continuous on [๐๐, ๐๐], AND the first derivative exists on the interval (๐๐, ๐๐) then there is at least one number ๐๐ = ๐๐ in (๐๐, ๐๐) such that ๐๐โฒ(๐๐) = ๐๐(๐๐)โ ๐๐(๐๐)
๐๐โ๐๐ .
Rolleโs Theorem
If the function ๐๐(๐๐) is continuous on [๐๐, ๐๐], AND the first derivative exists on the interval (๐๐, ๐๐) AND ๐๐(๐๐) = ๐๐(๐๐), then there is at least one number ๐๐ = ๐๐ in (๐๐, ๐๐) such that ๐๐โฒ(๐๐) = 0 .
Extreme Value Theorem
If the function ๐๐(๐๐) is continuous on [๐๐, ๐๐], then the function is guaranteed to have an absolute maximum and an absolute minimum on the interval.
Alternate Definition of the Derivative:
๐๐โฒ(๐๐) = lim๐๐ โ๐๐
๐๐(๐๐) โ ๐๐(๐๐)๐๐ โ ๐๐
Curve Sketching And Analysis
๐ฆ๐ฆ = ๐๐(๐๐) must be continuous at each:
Critical point: ๐๐๐๐๐๐๐๐
= 0 or undefined LOOK OUT FOR ENDPOINTS
Local minimum: ๐๐๐๐
๐๐๐๐ goes (โ, 0, +) or (โ,๐ข๐ข๐๐๐๐, +) OR ๐๐
2๐๐๐๐๐๐2
> 0 Local maximum: ๐๐๐๐
๐๐๐๐ goes (+, 0,โ) or (+,๐ข๐ข๐๐๐๐,โ) OR ๐๐
2๐๐๐๐๐๐2
< 0 Point of inflection: concavity changes ๐๐2๐๐๐๐๐๐2
goes from (+, 0,โ), (โ, 0, +), (+,๐ข๐ข๐๐๐๐,โ), OR (โ,๐ข๐ข๐๐๐๐, +)
Derivative of an Inverse Function: If f has an inverse function g then:
๐๐โฒ(๐๐) = 1
๐๐โฒ(๐๐(๐๐))
derivatives are reciprocal slopes
Implicit Differentiation
Remember that in implicit differentiation you will have a ๐๐๐๐
๐๐๐๐ for each y in the original
function or equation. Isolate the ๐๐๐๐๐๐๐๐
. If you
are taking the second derivative ๐๐2๐๐
๐๐๐๐2 , you
will often substitute the expression you found for the first derivative somewhere in the process.
Average Rate of Change ARoC:
๐๐๐ ๐ ๐ ๐ ๐๐ = ๐๐(๐๐) โ ๐๐(๐๐)
๐๐ โ ๐๐
Instantaneous Rate of Change IRoC:
๐๐๐ก๐ก๐๐๐๐ = ๐๐โฒ(๐๐) = limโโ0
๐๐(๐๐ + โ) โ ๐๐(๐๐)โ
Write the equation of a tangent line at a point:
You need a slope (derivative) and a point.
๐ฆ๐ฆ2 โ ๐ฆ๐ฆ1 = ๐๐ (๐๐2 โ ๐๐1)
First Derivative:
๐๐โฒ(๐๐) > 0 function is increasing.
๐๐โฒ(๐๐) < 0 function is decreasing.
๐๐โฒ(๐๐) = 0 or DNE: Critical Values at x.
Relative Maximum: ๐๐ โฒ(๐๐) = 0 or DNE and sign of ๐๐โฒ(๐๐) changes from + to โ .
Relative Minimum: ๐๐ โฒ(๐๐) = 0 or DNE and sign of ๐๐โฒ(๐๐) changes from โ to + .
Absolute Max or Min: MUST CHECK ENDPOINTS ALSO
The maximum value is a y-value.
Second Derivative:
๐๐โฒโฒ(๐๐) > 0 function is concave up.
๐๐โฒโฒ(๐๐) < 0 function is concave down.
๐๐โฒ(๐๐) = 0 and sign of ๐๐โฒโฒ(๐๐) changes, then there is a point of inflection at x. Relative Maximum: ๐๐โฒโฒ(๐๐) < 0 Relative Minimum: ๐๐โฒโฒ(๐๐) > 0
Horizontal Asymptotes:
1. If the largest exponent in the numerator is < largest exponent in the denominator then lim
๐๐โยฑโ๐๐(๐๐) = 0.
2. If the largest exponent in the numerator is > the largest exponent in the denominator then lim
๐๐โยฑโ๐๐(๐๐) = ๐ท๐ท๐ท๐ท๐ท๐ท
3. If the largest exponent in the numerator is = to the largest exponent in the denominator then the quotient of the leading coefficients is the asymptote. lim๐๐โยฑโ
๐๐(๐๐) = ๐๐๐๐
โPLUS A CONSTANTโ The Fundamental Theorem of Calculus
๏ฟฝ๐๐(๐๐)๐๐๐๐ = ๐น๐น(๐๐) โ ๐น๐น(๐๐)๐๐
๐๐
Where ๐น๐นโฒ(๐๐) = ๐๐(๐๐)
Corollary to FTC
๐๐๐๐๐๐
๏ฟฝ ๐๐(๐ก๐ก)๐๐๐ก๐ก =
๐๐(๐ข๐ข)
๐๐
๐๐๏ฟฝ๐๐(๐ข๐ข)๏ฟฝ ๐๐๐ข๐ข๐๐๐๐
Distance, Velocity, and Acceleration
๐๐(๐ก๐ก) = position function ๐ข๐ข(๐ก๐ก) = velocity function ๐๐(๐ก๐ก) =acceleration function
The derivative of position (ft) is velocity (ft/sec); the derivative of velocity (ft/sec) is acceleration (ft/sec2). The integral of acceleration (ft/sec2) is velocity (ft/sec) ; the integral of velocity (ft/sec) is position (ft).
Speed is | velocity |
If acceleration and velocity have the same sign, then the speed is increasing, particle is moving right.
If the acceleration and velocity have different signs, then the speed is decreasing, particle is moving left. Displacement = โซ ๐ข๐ข(๐ก๐ก) ๐๐๐ก๐ก๐ก๐ก๐๐
๐ก๐ก0
Distance = โซ | ๐ข๐ข(๐ก๐ก)|๐๐๐๐๐๐๐๐๐๐ ๐ก๐ก๐๐๐ก๐ก๐ ๐
๐๐๐๐๐๐๐ก๐ก๐๐๐๐๐๐ ๐ก๐ก๐๐๐ก๐ก๐ ๐ ๐๐๐ก๐ก Average Velocity
= final positionโ initial position
total time= ฮ๐๐ฮ๐ก๐ก
ONLY FOUR THINGS YOU CAN DO ON A CALCULATOR THAT NEEDS NO WORK SHOWN:
1. Graphing a function within an arbitrary view window.
2. Finding the zeros of a function. 3. Computing the derivative of a function numerically. 4. Computing the definite integral of a function
numerically.
LOGARITHMS Definition:
๐๐๐๐ ๐ท๐ท = ๐๐ โ ๐๐๐๐ = ๐ท๐ท ๐๐๐๐ ๐๐ = 1 ln 1 = 0 ln(๐๐๐ท๐ท) = ln๐๐ + ln๐ท๐ท
ln ๏ฟฝ๐๐๐ท๐ท๏ฟฝ = ln๐๐ โ ln๐ท๐ท
๐๐ โ ln๐๐ = ln๐๐๐๐
EXPONENTIAL GROWTH and DECAY: When you see these words use: ๐ฆ๐ฆ = ๐ถ๐ถ๐๐๐๐๐ก๐ก
โy is a differentiable function of t such that ๐ฆ๐ฆ > 0 and ๐ฆ๐ฆโฒ = ๐๐๐ฆ๐ฆ โ
โthe rate of change of y is proportional to yโ
When solving a differential equation:
1. Separate variables first
2. Integrate
3. Add +C to one side
4. Use initial conditions to find โCโ 5. Write the equation if the form of ๐ฆ๐ฆ = ๐๐(๐๐)
The Accumulation Function
๐น๐น(๐๐) = ๐๐(๐๐) + ๏ฟฝ๐๐โฒ(๐ก๐ก) ๐๐๐ก๐ก
๐๐
๐๐
The total amount, ๐น๐น(๐๐), at any time ๐๐ , is the initial amount, ๐๐ (๐๐), plus the amount of change between ๐ก๐ก = ๐๐ and ๐ก๐ก = ๐๐, given by the integral.
FOR MORE RESOURCES VISIT:
covenantchristian.org/bird/Calculus.htm
www.teacherspayteachers.com/Store/Jean-Adams
Mean Value Theorem for Integrals: The Average Value
If the function ๐๐(๐๐) is continuous on [๐๐, ๐๐] and the first derivative exists on the interval (๐๐, ๐๐), then there exists a number ๐๐ = ๐๐ on (๐๐, ๐๐) such that
๐๐๐๐๐๐๐๐ = 1
๐๐ โ ๐๐ ๏ฟฝ๐๐(๐๐) ๐๐๐๐ =
โซ ๐๐(๐๐)๐๐๐๐๐๐๐๐๐๐ โ ๐๐
๐๐
๐๐
This value ๐๐(๐๐) is the โaverage valueโ of the function on the interval [๐๐, ๐๐].
Riemann Sums
A Riemann Sum means a rectangular approximation. Approximation means that you DO NOT EVALUATE THE INTEGRAL; you add up the areas of the rectangles.
Trapezoidal Rule
For uneven intervals, may need to calculate area of one trapezoid at a time and total. ๐ด๐ด๐๐๐๐๐๐๐๐ = 1
2โ[๐๐1 + ๐๐2]
For even intervals:
๏ฟฝ๐๐(๐๐) ๐๐๐๐ ๐๐
๐๐
= ๐๐ โ ๐๐
2๐๐๏ฟฝ๐ฆ๐ฆ0 + 2๐ฆ๐ฆ1 + 2๐ฆ๐ฆ2 + โฆ
+2๐ฆ๐ฆ๐๐โ1 + ๐ฆ๐ฆ๐๐๏ฟฝ
Values of Trigonometric Functions for
Common Angles
๐๐ ๐ ๐ ๐ ๐ ๐๐ ๐๐ ๐๐๐๐๐ ๐ ๐๐ ๐ก๐ก๐๐๐๐ ๐๐
0 0 1 0
๐๐6
12
โ32
โ33
๐๐4
โ22
โ22
1
๐๐3
โ32
12
โ3
๐๐2
1 0 "โ"
๐๐ 0 โ1 0
Must know both inverse trig and trig values:
EX. ๐ก๐ก๐๐๐๐ ๐๐4
= 1 and sinโ1 ๏ฟฝ12๏ฟฝ = ๐๐
3
ODD and EVEN:
sin(โ๐๐) = โ sin ๐๐ (odd)
cos(โ๐๐) = cos ๐๐ (even)
Trigonometric Identities
Pythagorean Identities:
๐ ๐ ๐ ๐ ๐๐2๐๐ + ๐๐๐๐๐ ๐ 2๐๐ = 1
The other two are easy to derive by dividing by sin2 ๐๐ or cos2 ๐๐.
1 + tan2 ๐๐ = ๐ ๐ ๐๐๐๐2๐๐
cot2 ๐๐ + 1 = csc2 ๐๐
Double Angle Formulas:
sin 2๐๐ = 2 sin ๐๐ cos ๐๐
cos 2๐๐ = cos2 ๐๐ โ sin2 ๐๐ = 1 โ 2 sin2 ๐๐
Power-Reducing Formulas:
cos2 ๐๐ =12
(1 + cos 2๐๐ )
sin2 ๐๐ =12
(1 โ cos 2๐๐)
Quotient Identities: tan๐๐ = sin๐๐
cos๐๐ cot ๐๐ = cos๐๐
sin๐๐
Reciprocal Identities: ๐๐๐ ๐ ๐๐ ๐๐ = 1
๐ ๐ ๐๐๐๐ ๐๐ or sin ๐๐ csc ๐๐ = 1
๐ ๐ ๐๐๐๐ ๐๐ = 1cos๐๐
or cos ๐๐ sec๐๐ = 1
Area and Solids of Revolution:
NOTE: (๐๐,๐๐) are x-coordinates and (๐๐,๐ ๐ ) are y-coordinates
Area Between Two Curves: Slices โฅ to x-axis: ๐ด๐ด = โซ [๐๐(๐๐) โ ๐๐(๐๐)] ๐๐๐๐ ๐๐
๐๐
Slices โฅ to y-axis: ๐ด๐ด = โซ [๐๐(๐ฆ๐ฆ) โ ๐๐(๐ฆ๐ฆ)] ๐๐๐ฆ๐ฆ ๐๐๐๐
Volume By Disk Method:
About x-axis: ๐๐ = ๐๐ โซ [๐ ๐ (๐๐)]2 ๐๐๐๐ ๐๐๐๐
About y-axis: ๐๐ = ๐๐ โซ [๐ ๐ (๐ฆ๐ฆ)]2 ๐๐๐ฆ๐ฆ ๐๐๐๐
Volume By Washer Method:
About x-axis: ๐๐ = ๐๐ โซ ([๐ ๐ (๐๐)]2 โ [๐๐(๐๐)]2) ๐๐๐๐ ๐๐๐๐
About y-axis: ๐๐ = ๐๐ โซ ([๐ ๐ (๐ฆ๐ฆ)]2 โ [๐๐(๐ฆ๐ฆ)]2) ๐๐๐ฆ๐ฆ ๐๐๐๐
Volume By Shell Method:
About x-axis: ๐๐ = 2 ๐๐ โซ ๐ฆ๐ฆ [๐ ๐ (๐ฆ๐ฆ)] ๐๐๐ฆ๐ฆ๐๐๐๐
About y-axis: ๐๐ = 2 ๐๐ โซ ๐๐ [๐ ๐ (๐๐)] ๐๐๐๐๐๐๐๐
General Equations for Known Cross Section where base is the distance between the two
curves and a and b are the limits of integration.
SQUARES: ๐๐ = โซ (๐๐๐๐๐ ๐ ๐๐)2 ๐๐๐๐ ๐๐๐๐
TRIANGLES EQUILATERAL: ๐๐ = โ3
4 โซ (๐๐๐๐๐ ๐ ๐๐)2 ๐๐๐๐ ๐๐๐๐
ISOSCELES RIGHT: ๐๐ = 14 โซ (๐๐๐๐๐ ๐ ๐๐)2 ๐๐๐๐ ๐๐
๐๐
RECTANGLES: ๐๐ = โซ (๐๐๐๐๐ ๐ ๐๐) โ โ ๐๐๐๐ ๐๐๐๐
where h is the height of the rectangles.
SEMI-CIRCLES: ๐๐ = ๐๐2 โซ (๐๐๐๐๐๐๐ ๐ ๐ข๐ข๐ ๐ )2 ๐๐๐๐ ๐๐
๐๐ where radius is ยฝ distance between the two curves.
Basic Integrals
๏ฟฝ๐๐๐ข๐ข = ๐ข๐ข + ๐ถ๐ถ
๏ฟฝ๐ข๐ข๐๐๐๐๐ข๐ข = ๐ข๐ข๐๐+1 ๐๐ + 1
+ ๐ถ๐ถ ๐๐ โ โ1
๏ฟฝ๐๐๐ข๐ข๐ข๐ข
= ln| ๐ข๐ข | + ๐ถ๐ถ
๏ฟฝ๐๐๐ข๐ข ๐๐๐ข๐ข = ๐๐๐ข๐ข + ๐ถ๐ถ
๏ฟฝ๐๐๐ข๐ข ๐๐๐ข๐ข = ๐๐๐ข๐ข
ln ๐๐ + ๐ถ๐ถ
๏ฟฝ sin๐ข๐ข ๐๐๐ข๐ข = โ cos๐ข๐ข + ๐ถ๐ถ
๏ฟฝ cos๐ข๐ข ๐๐๐ข๐ข = sin๐ข๐ข + ๐ถ๐ถ
๏ฟฝ tan๐ข๐ข ๐๐๐ข๐ข = โ ln|cos๐ข๐ข + ๐ถ๐ถ|
๏ฟฝ cot๐ข๐ข ๐๐๐ข๐ข = ln|sin๐ข๐ข| + ๐ถ๐ถ
๏ฟฝ sec๐ข๐ข ๐๐๐ข๐ข = ln|sec๐ข๐ข + tan๐ข๐ข| + ๐ถ๐ถ
๏ฟฝ csc๐ข๐ข ๐๐๐ข๐ข = โ ln|csc๐ข๐ข + cot๐ข๐ข| + ๐ถ๐ถ
๏ฟฝ sec2 ๐ข๐ข ๐๐๐ข๐ข๐๐ = tan๐ข๐ข + ๐ถ๐ถ
๏ฟฝ csc2 ๐ข๐ข ๐๐๐ข๐ข = โ cot๐ข๐ข + ๐ถ๐ถ
๏ฟฝ sec๐ข๐ข tan๐ข๐ข ๐๐๐ข๐ข = sec๐ข๐ข + ๐ถ๐ถ
๏ฟฝ csc๐ข๐ข cot๐ข๐ข ๐๐๐ข๐ข = โ csc๐ข๐ข + ๐ถ๐ถ
๐๐ = ๐๐ ๐๐ = ๐๐๐๐ ๐๐ = ๐๐๐๐ ๐๐ = |๐๐|
๐๐ = โ๐๐ ๐๐ =๐๐๐๐
๐๐ = ๐๐๐๐๐๐ ๐๐ ๐๐ = ๐๐๐๐๐๐ ๐๐
๐๐ = ๐๐๐๐ ๐๐ = ๐๐๐๐ ๐๐ ๐๐ =๐๐๐๐๐๐
๐๐ = ๏ฟฝ๐๐๐๐ โ ๐๐๐๐
MORE DERIVATIVES:
๐๐๐๐๐๐๏ฟฝsinโ1 ๐ข๐ข
๐๐๏ฟฝ = 1
โ๐๐2โ ๐ข๐ข2 ๐๐๐ข๐ข๐๐๐๐
๐๐๐๐๐๐
[cosโ1 ๐๐] = โ1โ1โ ๐๐2
๐๐๐๐๐๐๏ฟฝtanโ1 ๐ข๐ข
๐๐๏ฟฝ = ๐๐
๐๐2+๐ข๐ข2 ๐๐๐ข๐ข๐๐๐๐
๐๐๐๐๐๐
[cotโ1 ๐๐] = โ11+๐๐2
๐๐๐๐๐๐๏ฟฝsecโ1 ๐ข๐ข
๐๐๏ฟฝ = ๐๐
|๐ข๐ข|โ๐ข๐ข2โ ๐๐2 ๐๐๐ข๐ข๐๐๐๐
๐๐๐๐๐๐
[cscโ1 ๐๐] = โ1|๐๐|โ๐๐2โ 1
๐๐๐๐๐๐
(๐๐๐ข๐ข) = ๐๐๐ข๐ข ln๐๐ ๐๐๐ข๐ข๐๐๐๐
๐๐๐๐๐๐
[log๐๐ ๐๐] = 1๐๐ ln ๐๐
MORE INTEGRALS:
๏ฟฝ๐๐๐ข๐ข
โ๐๐2 โ ๐ข๐ข2= sinโ1
๐ข๐ข๐๐
+ ๐ถ๐ถ
๏ฟฝ๐๐๐ข๐ข
๐๐2 + ๐ข๐ข2=
1๐๐
tanโ1๐ข๐ข๐๐
+ ๐ถ๐ถ
๏ฟฝ๐๐๐ข๐ข
๐ข๐ข โ๐ข๐ข2 โ ๐๐2=
1๐๐
secโ1|๐ข๐ข|๐๐
+ ๐ถ๐ถ
โ1 1
โ1
1
x
y
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