ap concepts sheet

8/6/2019 AP Concepts Sheet

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AP Exam Concepts & Key Word List

Calculus Concept Definition or Hint Continuous lim ( ) lim ( ) ( )

x a x a f x f x f a

No holes, asymptotes, breaksDifferentiable Derivative from the left and right have the same

value.F(x) is continuousThe graph is a SMOOTH curve.No sharp turns, vertical tangent lines, etc.

Equation of a tangent line Need Slope and a coordinateSlope = f (c)At the given coordinate the tangent line AND thefunction have the same value.

Linear Approximation Use an equation line to approximate the y-value onthe function. (Because the gap between thetangent and the function is small)

Limit Definition of a Derivative ( ) ( )lim '( )

x x

f x x f x f x

x

( ) ( )lim '( ) x c

f x f c f c

x c; derivative evaluated at

x=cEquation of a normal line This line is perpendicular to the tangent line.

Find slope using f (c) but then take negativereciprocal of that value.

Equation of a secant line Equation of a line between 2 points on a curve.

Critical Value Where f (x) = 0 or undefinedF(x) =0 POSSIBLE max/min for f(x)

F(x) has horizontal tangent lineF(x) touches or crosses the x-axis

F is Increasing F(x) >0When y values in a table are increasing

F is Decreasing F(x) 0When the first derivative is increasing b/c tangentlines to first derivative will be positive

F is Concave down F(x)


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Absolute Extrema Occurs where f (x) = 0 or undefined or at endpointsof a closed interval.Highest y-value is abs maxLowest y-value is abs min

Minimum or Maximum Value Refers to Y-VALUES!!

Minimum is 5 means (x,5)

F(x) =0 POSSIBLE inflection pointsWhere the 1 st derivative has horizontal tangentlines.Where the 2 nd derivative graph touches or crossesthe x-axis

Inflection point Where f (x)=0 or undefined AND the 2 nd derivativehas a sign change

Horizontal Asymptote lim ( ) _ x

f x numeric value

lim ( ) _ x

f x numeric value

Y = # is the horizontal asymptoteUse your short cut rules if possible

Removable Discontinuity When you reduce a fraction by a common factorSet common factor = 0 and solve to obtain locationof the discontinuity.Hole in the graph

Vertical Asymptote After you have factored and reduced a fraction,where the denominator = 0X = # is the vertical asymptote

Zero Set equation equal to 0Graphically, where function crosses the x-axis Average Rate (of change) ( ) ( )

'( )f b f a

f cb a

RATE = derivative Average Number 1

( )b

a

f x dxb a

Rolle s Theorem Function must be continuous on [a,b] anddifferentiable on (a,b) and f(a) = f(b). Thenguaranteed c value where f (c) = 0 Guarantees c value where f has a horizontaltangent line.

Intermediate Value Theorem F(x) is continuous on the closed interval [a,b] and kis a number between f(a) and f(b), then there is a cvalue in [a,b] such that f(c) = k

Can be used to prove existence of a zeroCan be used to prove existence of a value


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Mean Value Theorem Function must be continuous on [a,b] anddifferentiable on (a,b).

( ) ( )'( )

f b f a f c

b a

Where c must be INSIDE the interval given

Guarantees a c value where the secant line isparallel to the tangent line.

Position Original Function, x(t)Height Measurement: ft, m

Velocity First derivative of PositionSpeed with directionNegative value means moving down or leftPositive value means moving up or rightX(t) = v(t)Measured in ft/sec or m/sec

Speed The absolute value of velocity

( )v t Measured in ft/sec or m/sec

Acceleration 2nd Derivative of PositionFirst derivative of VelocityX(t) = v (t) = a(t)Measured in ft/sec 2 or m/sec 2

Total Distance or Total Amount ( ) '( )v t dt f x dx rate dt

Integrate the absolute value of velocity/rateOr separate this into multi-integrals based onwhere the function crosses the x-axis.

We must compensate for any regions that fallbelow the x-axis (we need to count them as +)

Maximize --?-- (volume, area, velocity, etc) To maximize something, you need to use the 1 st derivative, find critical values, test them todetermine max (you may use 2 nd derivative test)

Minimize --? (volume, area, etc) To minimize something, you need to use the 1 st derivative, find critical values, test them todetermine min (you may use 2 nd derivative test)

Riemann Sum Summing up areas of rectangles.Rectangles can be equal width: (b-a)/nRectangles can be varied widths: calculate thewidth of each by hand; use a number line diagramto help.

Inscribed Rectangles Every rectangle falls BELOW the function.Use the graph to determine whether you shoulduse left or right methods. It is possible you mayhave to use a combination of both.


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Circumscribed Rectangles Every rectangle falls ABOVE the function.Use the graph to determine whether you shoulduse left or right methods. It is possible you mayhave to use a combination of both.

Midpoint Sum The height falls in the middle of the rectangle.Left Sum The height falls in on the left side of the rectangle.The left side of the rectangle touches the function.

Right Sum The height falls in on the right side of the rectangle.The right side of the rectangle touches the function.

Trapezoidal SumFixed width is:

2

b an

Coefficients are: 1 2 2 2 2 1

Area of a trapezoid is: 1 2

2

b bh

Using area to find the value of an integral Area is always positiveIntegral values can be negative.If you use area to help you evaluate an integral,regions above the x-axis are considered positivevalues. Regions below the x-axis are considerednegative values.

Derivative of an integral( )

( ) ( ( )) '( )g x

a

d f t dt f g x g x

dx

2nd

Fundamental Theorem of CalculusThe rate of change of y with respect to timeis (directly) proportional to y.

dyky

dt kt y Ce

The rate of change of w with respect to timeis proportional to w . (or any variable-must be same)

dwkw

dt kt W Ce

The rate of change of y with respect to timeis jointly proportional to t ( )

dyk yt

dt

The rate of change of y with respect to timeis inversely proportional to t

dy yk

dt t

'( )b

a

f x dx F(b) F(a)Integrate; plug in the top and plug in the bottom

Limit Definition of an Integral

1

( ) lim ( )b

inia

f x dx f c x

Try to pick out the f(x)Use the c i to help you find the a valueUse the deltax and the a value to find the b


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Area Set up the integral based on the region describedCheck for intersection point(s)Check for top bottom

( ) ( )b

a

top bottom dx

Volume of region R created with crosssections perpendicular to the x-axis

Integrate the AREA of the shape described.

( )b

a

A x dx

Volume of regionDisc Method: 2 ( )

b

a

R x dx

Washer Method: 2 2( ) ( )b

a

R x r x dx

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