2Nd Semester Calculus Vocab

Front 1

Critical Value

Back 1

An x-value that makes f'(x)=0 or f'(x) undefined.

Front 2

Extrema

Back 2

The max and min of a function.

Front 3

The First Derivative Test

Back 3

This test helps us identify if a critical point is a max or min. To use the test, first find the critical values; then graph them on a number line and test values in between the C.P by plugging them in to f' and observing the sign. When the derivative changes from + to - at a point that point is a max. When the derivative changes from - to + its a min.

Front 4

Interval of increase

Back 4

The part of a graph with positive slopes. Analytically this happens on interval where f'(x)>0

Front 5

Interval of decrease

Back 5

The part of a graph with negative slopes. Analytically this happens on intervals where f'(x)<0

Front 6

Inflection Point

Back 6

A point where the concavity of a function changes. Analytically, this happens at the x-values for which f''(x)=0.

Front 7

Concave Up

Back 7

The part of a graph where it is curved upwards. Analytically, this happens on intervals where f''(x)>0.

Front 8

Concave Down

Back 8

The part of a graph where it is curved downwards. Analytically, this happens on intervals where f''(x)<0.

Front 9

Second Derivative Test.

Back 9

This is a test using the second derivative to deterring whether a critical point is a max or a min. To use this test first find the critical values then test them in the second derivative:

If f'(a)=0 and f''(a) is positive then f(x) has a local min at x=a.

If f'(a)=0 and f''(a) is negative then f(x) has a local max at x=a.

If f'(a)=0 and f''(a)=0 or DNE then a is neither a min nor a max.

Front 10

Velocity Function

Back 10

First Derivative of the Position Function.

Front 11

Acceleration Function

Back 11

Second Derivative of the Position Function (and first derivative of velocity).

Front 12

Area of a Rectangle

Back 12

A=b*h

Front 13

Area of a Triangle

Back 13

A= 1/2 b * h

Front 14

Area of a Circle

Back 14

A=pi * r^2

Front 15

Circumference of a Circle

Back 15

C= 2 * pi * r

Front 16

Volume of a box

Back 16

V= b * h * w

Front 17

Volume of a Prism

Back 17

V= B * h ( B is the area of the base which can be different shapes)

Front 18

Volume of a cone/pyramid

Back 18

V= 1/3 * B * h

Front 19

Optimization

Back 19

The process of finding an ideal condition (i.e. maximizing or minimizing something).

Front 20

Locally Linear (at a point)

Back 20

A function is locally linear at a point if, when we zoom in on the function the curve starts to look like a straight line. (If the function is "locally linear" it is differentiable).

Front 21

Linearization

Back 21

A linear equation whose y-values can be used to approximate a function near a particular x-value. (essentially it is the equation of a tangent line).

Front 22

Newton's Method (say the formula and the idea!)

Back 22

IDEA: A process for estimating the root of a function using tangent lines and their x-intercepts.

FORMULA: new x = old x - f(old x)/ f'(old x)

Front 23

When does Newton's Method fail?

Back 23

1) When we choose an x-value with a slope of 0

2) When we start too far from the root we are trying to find.

Front 24

Differential (what is it?)

Back 24

A variable that represents a small change.

Front 25

What does dy equal? (differential formula)

Back 25

dy= f'(x) * dx

Front 26

What is the difference between absolute and relative change? How do you calculate them?

Back 26

Absolute Change= the difference in the numbers (subtract the outputs before and after)

Relative Change= the percent that changed

(divide Absolute Change by the original output)

Front 27

What is the the meaning of:

Back 27

The area under the f curve between a and b.

Front 28

What is "RAM"? How do you set it up? What determines its accuracy?

Back 28

RAM is the "Rectangular Approximation Method." It approximates the area under a curve by dividing that area in to a bunch of rectangles. To set it up you simply multiply the width (delta x) times the height (y-value) of each rectangle. The accuracy depends on how many rectangles you use (n)... more is better.

Front 29

What is "LRAM"? How does its set up differ from RRAM? How do you know if it is an over or underestimate?

Back 29

The "Left Hand Approximation Method" is a type of RAM where we use the y-values (heights) that correspond to the left edge of each rectangle.

LRAM is an underestimate when the function is increasing. LRAm is an overestimate when the function is decreasing.

Front 30

What is "RRAM"? How does its set up differ from LRAM? How do you know if it is an over or underestimate?

Back 30

The "Right Hand Approximation Method" is a type of RAM where we use the y-values (heights) that correspond to the right edge of each rectangle.

RRAM is an overestimate when the function is increasing. RRAm is an underestimate when the function is decreasing.

Front 31

What is the Trapezoidal Approximation Method? How do you know if it is an over or under estimate?

Back 31

The Trapezoidal Approximation method allows us to estimate the area under a curve by dividing it up in to trapezoids.

It is an overestimate when the function is concave up and an underestimate when the function is concave down.

Front 32

What is the formula for the area of a trapezoid?

How does it relate/compare to the shortcut formula for the trapezoidal approximation method?

Back 32

Area of a trapezoid: width * (height1 + height2)/2

Trapezoidal Approximation Method Shortcut:

(only works if the widths are all the same!)

width*(height1 + 2* height2 + 2* ... +height)

Front 33

In applied problems what does the area under the rate curve represent?

Back 33

The total change in the original function from the start to the end of that interval.

Front 34

According to the first part of the FTC, what does the expression below equal?

Back 34

f(b) - f(a)

Front 35

According to the second part of the FTC, what is the derivative of the function below:

Back 35

f(x)

Front 36

What does " anti-derivative of f(x) " mean?

Back 36

A function whose derivative is f(x)... every function has a family anti-derivatives which differ only by a constant value ( + C).

Front 37

When Newton and Leibniz were inventing Calculus, how did they use the idea of RAM to come up with a precise way to calculate the area under a curve?

Back 37

They took the limit of RRAM as n approached infinity... thus they made their approximations infinitely close to the actual value of the area under the curve.

Front 38

What is the formula for the average value of a function?

Back 38

1/ (b-a ) * integral from a to b of f(x)dx.

Front 39

What are the two situations in which we consider the "area under the curve" to be negative?

Back 39

1) regions below the x-axis

2) when the limits of integration are reversed (i.e. if the upper limit is lower then the lower limit... i.e. going from right to left)

Front 40

What is the area under the sin curve from 0 to pi?

Back 40

2 units ^2

Front 41

What is a differential equation? What are the two kinds of solutions to a differential equation?

Back 41

A differential equation is an equation with a term that represents a derivative. Its solution is a function for the original quantity.

GENERAL SOLUTION - represents a family of functions ( and includes + C)

PARTICULAR SOLUTION - represents and exact function and requires more information so that we can solve for C.

Front 42

What is a slope field, when are they helpful?

Back 42

A slope field is a picture of how a function's shape depends on x and y. They are useful for getting a sense for what an anti-derivative might look like particularly in cases where we can't find one algebraically.

Front 43

What is the integration by parts formula?

Back 43

uv - int( v du)

Front 44

State the 3 pythagorean trig identities.

Back 44

sin^2 + cos^2 = 1

tan^2 = sin^2-1

cot^2 = csc^2-1

Front 45

State the two situations when a definite integral will equal zero.

Back 45

1) when the upper and lower limits of integration are the same number

2) when the area above the x-axis equals the area under the x-axis.

Front 46

State the two power reducing identities.

(cos^2 = and sin^2 = )

Back 46

cos^2(x) = (1+cos(2x))/2

sin^2(x) = (1-cos(2x))/2

Front 47

If you are rotating an area around the the x-axis and using dx as your variable of integration, what method should you use?

Back 47

Washers/Disks

Front 48

If you are rotating an area around the the x-axis and using dy as your variable of integration, what method should you use?

Back 48

Shells

Front 49

If you are rotating an area around the the y-axis and using dx as your variable of integration, what method should you use?

Back 49

Shells

Front 50

If you are rotating an area around the the y-axis and using dy as your variable of integration, what method should you use?

Back 50

Washers/Disks

Front 51

What formula will help you calculate the length of a curve?

Back 51

integral of SQRT ( 1 + (f'(x))^2)dx

Front 52

If you are calculating the area between two curves and want to use horizontally oriented rectangles... what should your variable of integration be?

Back 52

dx

Front 53

If you are calculating the area between two curves and want to use vertically oriented rectangles... what should your variable of integration be?

Back 53

dy