Term
What will be on the final concerning Limits?
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Definition
Finding Limits Using Factoring
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Finding Limits Using Limit Formula
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Finding Limits When X approaches Infinity
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Term
How Do You Find Instantaneous Rate of Change and Find The Equation of The Tangent Line?
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Definition
1.) Take the Derivative of the Function
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2.) Plug the value of "x" into the derivative to solve for slope
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3.) Plug the value of "x" back into the original function, to solve for "y1"
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4.) x=x1, so you don't need to solve for x1
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5.) You now know x1, y1, and the slope.
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6.) Now plug those values into the point slope formula to solve for the equation of the tangent line.
y-y1=m(x-x1)
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Term
How to Solve for Instantaneous Rate of Change (Slope)?
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Definition
1.) Take the Derivative of the Function
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2.) Plug the value of "x" into the derivative to solve for slope (Instantaneous Rate of Change)
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Term
What is the Derivative?
(Ax^B)+(Cx^D)+(Ex^F)
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Definition
(BAx^(B-1))+(DCx^(D-1))+(EFx^(F-1))
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Term
What is the Negative Exponent Rule?
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Definition
A^-B = 1/A^B
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1/A^-B = A^-B/1
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Term
What is The Solution?
A^0
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Definition
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Term
What is the Derivative of?
Ax^B
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Definition
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Term
Dividing Derivatives
A(x)/B(x)
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Definition
((A'(x)*B(x))-(B'(x)*A(x)))/(B(x)^2)
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Term
Multiplying Derivatives
A(x)*B(x)
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Definition
(A'(x)*B(x))+(B'(x)*A(x))
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Term
Average Rate of Change
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Definition
(f(x1)-f(x2))/(x1-x2)
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Term
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Definition
log x + log y
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Term
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Definition
log x - log y
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Term
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Definition
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Term
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Definition
ln x / ln10
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Term
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Definition
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Term
How do you solve the following?
lna'
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Definition
Treat "a" as if it is a function
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Find a'
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Set a'/a
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Your solution is
a'/a
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Term
If y = ln Ax
What does y' =
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Definition
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Term
If y = ln Ax+B
What does y' =
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Definition
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Term
A(Bx^C)^D' = ?
[If you understand this, you understand Chain Rule]
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Definition
(A*D(Bx^C)^(D-1))*((BX^C)')
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Term
If y = e^x
Then
y' = ?
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Definition
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Term
If
y = e^g(x)
Then
y' = ?
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Definition
(g(x)')e^(g(x))
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Term
How do you do a first derivative test (find the behavior of a function analytically without plotting it)?
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Definition
1.) Take the Derivative of the Function
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2.) Set the Derivative Function to Equal Zero and solve it for "x"
2a.) Factor it further to where you have multiple values of "x"
or
2b.) Use the Quadratic Formula to Obtain Multiple Values For "x"
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3.) Use the multiple values of "x" you have found, and set a test value between -infinity and your first value, your first value and your second value, and your second value and positive infinity. Represent this information as a chart.
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4.) Substitute your test values into the derivative in place of "x" and mark whether the solution to each value is positive or negative and whether the function is increasing or decreasing on your chart
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5.) Your x-values are the critical points, mark whether they are maximums or minimums depending on whether or not the slope right before the function is positive or negative
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Term
How do you do a second derivative test (find the behavior of a function analytically without plotting it)?
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Definition
1.) Take the first derivative of the function
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2.) Set the Derivative Equal to Zero and factor it
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3.)Set each factor equal to zero, and solve for critical points
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4.)Take the Second Derivative of the Original Function
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5.) Plug each critical point into the second derivative in place of "x", and record whether each one yields a positive solution or a negative solution. If a solution is negative, then that critical point is a maximum. If a solution is positive, then that critical point is a minimum.
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Term
How do you find the point of infliction (aka the point of diminishing returns)?
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Definition
1.)f(x)''
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2.)Set f(x)''=0
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3.)Solve f(x)''=0 for "x"
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Term
How do you sketch the curve of a function?
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Definition
1.) Find the x and y intercepts of the function
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2.) Find the Behavior of the function Using Either the First or Second Derivative Test: which will tell you of a curve's local maximum and local minimum
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3.) Use the information you have gathered from the first two steps, to sketch the curve
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Optional:
You can find the points of infliction, and graph those as well.
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Term
If the second derivative test yields only a zero, and zero is not positive or negative, then...
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Definition
The second derivative test has failed
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Term
What is a point of infliction?
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Definition
A point in which a function changes concavity
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Term
How do you find an antiderivative?
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Definition
1.)If it is in this format: x^ndx
Use the Power Rule of Antiderivatives
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2.) If it is in this format:
Ax^bdx=?
You will do the following:
A(1/b)x^(b+1)+C
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3.) If it is in this format:
(Ax^2 - Bx + C)dx = ?
You will do the following:
Ax^2dx - Bxdx + Cdx
and use this edited Power Rule of Antiderivatives on Each Individual Part
x^ndx = (1/n+1)x^(n+1)
Where n cannot equal -1
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4.) If it is 1dx your solution is:
x+C
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5.) If x^-ndx:
ln|x| + C or 1/x+C
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Term
What is the Power Rule of Antiderivatives?
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Definition
x^ndx = (1/n+1)x^(n+1)+C
Where n cannot equal -1
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Term
What is the Antiderivative Rule of Exponential Functions?
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Definition
1.) if:
e^xdx = e^x + C
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2.) If:
Ae^xdx = (1/Ae^(Ax))+C
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Term
If you have a function with three variables and you know the values of two of the variables, how do you determine the value of the third variable?
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Definition
Plug in the two values you know, and solve for the unknown variable.
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Term
How do you find the area between two curves within an interval of (a,b) on the x-axis?
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Definition
Definite integral of the first function minus the definite integral of the second function.
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Term
How do you take the definitive integral of a function, when trying to find the area within the integral of a to b?
[a is always the smaller number]
[the integral is always defined on the x-axis]
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Definition
1.) Find the antiderivative of f(x)
[Don't add the +C at the end]
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2.) Your function is now:
f(x) = the antiderivative of f(x)
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3.) Find f(a)
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4.) Find f(b)
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5.) f(b)-f(a)
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