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By Jeff Willets

AP Calculus Review

Many slides will send you to websites for additional examples. A very good website is “Visual Calculus” at http://archives.math.utk.edu/visual.calculus/index.html. Click anywhere in this box to go there.

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• Mean Value Theorem• Max/Mins—Points Of Inflection• Fundamental Theorems of Calculus• Reimann Sums• Trapezoid Rule• Linear Approximations• Motion Problems—position, velocity, acceleration• Solids of Rotation• List of Websites

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In other words, there must be a point somewhere on the curve with a tangent line parallel to the line connecting the endpoints.

If f(x) is a differentiable function on (a, b) , then there exists a value c on (a, b) such that

abafbfcf

)()()(

The Mean Value Theorem

Examples—Click on graphs

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First Derivatives

Consider a function f(x).

• Whenever f’(x)>0, f(x) is increasing• Whenever f’(x)<0, f(x) is decreasing

If f(c) is defined and f’(c) = 0 or f’(c) is undefined, c is called a critical value. (Caution: There are cases where f’(c) is undefined but so is f(c). This would not be a critical value. An example of this would be the function y = 1/x.)

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First Derivatives f’(c) is the slope of the tangent line at c.Assuming f(c) is defined, if:• f’(c)=0, there is a horizontal tangent line (Points A and B)• f’(c) is undefined, there is a vertical tangent line (C and D)

BA C D

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If c is a critical value, then f(c) could be a relative maximum, relative minimum, or neither. We can determine this with a first derivative test.

If f’(x) is negative to the left and positive to the right of c, then f(c) is a relative minimum.If f’(x) is positive to the left and negative to the right of c, then f(c) is a relative maximum.If f’(x) is positive on both sides of c or negative on both sides of c, then f(c) is neither a minimum or a maximum.

Max/Mins

--- +++c

+++ +++c

--- ---c

+++ ---c

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Endpoints on a closed interval must always be considered for absolute max/mins.

To determine absolute max/mins, first you must make sure they exist. If any limits are +/- infinity (at asymptotes or extremes) appropriate absolutes will not exist.

If absolutes do exist, they will be the endpoint or relative max/mins with the greatest (or least) y-values.

Max/Mins

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The second derivative tells you the concavity of the function.

• If f’’(x)>0, the function is concave up.• If f’’(x)<0, the function is concave down.

A point of inflection iswhere the functionswitches concavity (A, B, and C)

Second Derivatives/Points of Inflection

AB

C

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If f(c) exists and f’’(c)=0 or f’’(c) is undefined, then (c, f(c)) is a possible point of inflection.

• If f’’(x) switches from positive to negative or negative to positive at c, then (c, f(c)) is a point of inflection.

• If f’’(x) does not switch signs at c, then (c, f(c)) is not a point of inflection.

Second Derivatives/Points of Inflection

--- +++c

+++ ---c

--- ---c

+++ +++c Table of

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The second derivative test can also be used to determine if critical points are maximums or minimums.

So if f(c) is defined and f’(c)=0 or f’(c) is undefined and:

• f’’(c)>0, then (c, f(c)) is a relative minimum (since the function is concave up at that point)

• f’’(c)<0, then (c, f(c)) is a relative maximum (since the function is concave down at that point)

• f’’(c)=0, then the 2nd derivative test is inconclusive. A first derivative test must be used in this case.

Second Derivatives/Tests for Max/Mins

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f is a continuous function on [a, b] and F’=f.

Then

There is also the Second Fundamental Theorem. It states that

Fundamental Theorem

( ) ( )b

af x dx F b F a

( ) ( )x

a

d f t dt f xdx

Examples (atBottom of site)

Examples

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Riemann SumsHere are the three standard types of Riemann Sums, each broken into ten rectangles

Left-handed Riemann Sum—Notation L(10)

Midpoint Riemann Sum—Notation M(10)

Right-handed Riemann Sum—Notation R(10)

Visit Website withmore explanationsAnd examples.

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Trapezoid Rule

T(10).

12 0 1 2 12 2 ... 2n n nT x y y y y y

n is the number of trapezoids, Δx is the width of each trapezoid (which can be determined by (b-a)/n.) Note that there will always be one more term in the parentheses than there are trapezoids.

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Linear ApproximationsLinear Approximation is a method to approximate a value by using a value along the tangent line close to the point of tangency. If (a, f(a)) is the point of tangency of the line to the function f(x) below, then for x values “near” a,

This is merely the point-slope equation of line.

f x f a x a f a

(x,f(x))(a,f(a))

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Motion ProblemsLet s(t) be the position of an object at time t. (Sometimes it might be called x(t) or y(t).)

Then v(t) = s’(t), the velocity at time t.a(t) = v’(t) = s’’(t), the acceleration at time t.The sign of the velocity tells the direction it is moving. Positive usually means right or up.The sign of the acceleration tells the direction it is accelerating.

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Motion ProblemsSpeed—the difference between speed and velocity is that velocity has direction, and speed does not.Whenever the velocity and acceleration have the same sign, speed is increasing. When they have different signs, the speed is decreasing.

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Motion ProblemsFinding PositionOften you will be given a velocity function v(x) and an initial position s(0). You can find a position s(t) by:

0

( ) (0) ( )t

s t s v x dx This follows directly from the fundamental theorem of calculus.

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Motion Problems

Total Distance Traveled vs. DisplacementDisplacement is the net change in position (final position – starting position) It is found as follows:

0

( )ft

tv x dx

Total distance traveled is found as follows:

0

( )ft

t

v x dx

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Solids of RotationThere are basically two types of rotations, the disc/washer method and the shell method. The main difference is that the disc/washer method has the rectangles sliced perpendicular to the axis of rotation, and the shell method has them sliced parallel to the axis of rotation. Whenever the rectangles are vertical, the variable will be x (and dx) and the limits of integration will be the x limits. When the rectangles are horizontal, the variable will be y (and dy) and the limits of integration will be the y limits.

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Disc MethodIn this example, we are going to rotate the region bounded by y = x2, y=1, and the y axis about the y-axis. Slicing it horizontally, we get rectangles perpendicular to the axis of rotation, which is what is needed for the disc/washer method. We will use the formula 2

1

2y

yR dy

R is the distance from the axis of rotation to the function (note that we must convert everything to be in terms of y.) R = y1/2

12

1 2

0y dy

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Washer MethodIn this example, we are going to rotate the region bounded by y = x and y = x4 about the line y = 2. Slicing it vertically, we get rectangles perpendicular to the axis of rotation, which is what is needed for the disc/washer method. We will use the formula

2

1

2 2x

xR r dx

R and r are the distances from the axis of rotation to the functions (with R being the bigger one.) In this example, R = 2-x4 and r = 2-x.

1 2 24

02 2x x dx

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Shell MethodIn this example, we are going to rotate the region bounded by y = x and y = x4 about the line x = 1. Slicing it vertically, we get rectangles parallel to the axis of rotation, which is what is needed for the shell method. We will use the formula 2

1

2x

xrhdx

r is the distance from the axis of rotation to the rectangle, and h is the height of the rectangle. In this example, r = 1-x and h = x-x4

1

4

02 1 x x x dx

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Known Cross SectionsWith known cross sections, you will be given a base and told that the base will be the base of certain kind of shapes. In this example, the base will be the area between y = x4 and y = x, and each rectangle will be the base of a square. We will always use the formula 2

1

x

xAdx

A is the area of the square, which will be h2, or (x – x4)2.

1 24

0x x dx

Visit Website withmore explanationsAnd examples.

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List of Websites archives.math.utk.edu/visual.calculus

/ http://mathdemos.gcsu.edu/mathdem

os/solids/index.html www.sosmath.com/calculus/calculus.h

tml www.plu.edu/~heathdj/java/ people.hofstra.edu/faculty/

Stefan_Waner/RealWorld/tccalcp.html apcentral.collegeboard.com/ Calculus The Musical I Will Derive