Section 1
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Reciprocal Rule
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Section 1
(50 cards)
Reciprocal Rule
Function 1/f Derivative −f'/f^2
When is the second derivative of a function negative?
When the graph of the function is concave down
When is the second derivative of a function positive?
When the graph of the function is concave up
Derivative of e^x
e^x times derivative of x
What are the derivatives of trig functions?
sin(x) = cos (x); cos (x) = -sin(x); tan(x) = sec^2(x)
When can removable discontinuities be fixed?
Removable discontinuities can be "fixed" by re-defining the function.
Mean Value Theorem
F'(c)= f(b)-f(a)/b-a
f^-1 represents what?
An inverse function
Second Derivative
Take first derivative. Then, find the derivative of the first derivative. f'(x), then f''(x).
How do you find the absolute extrema of a function? How can you find the absolute extrema of a function on an interval with end points?
Find critical points by funding where the first derivative is 0 or undefined, then plug in end points to f(x) and critical points to find extrema.
What is point-slope form?
Increasing Functions
Where the graph of the first derivative shows the original function being continuous, differentiable and increasing.
What is an inflection point?
A point of a curve at which a change in the direction of curvature occurs.
How do you find the limit of a piece-wise function?
Step 1 Evaluate the one-sided limits for each function. Step 2 If the one-sided limits are the same, the limit exists. If the one-sided limits are different, the limit doesn't exist.
How do you find a Local Extrema?
1. Find the first derivative of f using the power rule. 2. Set the derivative equal to zero and solve for x. These x-values are the critical numbers of f. create intervals around the critical numbers to test f' to see if f is increasing or decreasing on either side of the critical numbers.
Power Rule
Function - x^n Derivative - 〖n〗x^(n-1)
If the appropriate conditions are satisfied, what does the Mean Value Theorem guarantee?
There is at least one point c in the interval (a, b) at which f'(c) = [f(b) - f(a)] / [b - a]
Derivative of tangent inverse of x
What are the 1st and 2nd derivatives of displacement?
1st derivative is velocity and the 2nd is acceleration. These are found by identifying the slope of displacement to find velocity, and slope of velocity to find acceleration
What are discontinuities? When are limits nonexistent?
Limits dont exist when the values from the left and righ are3 no equal
When is a function decreasing?
When the first derivative/ slope is negative
Chain Rule (Using ' )
Function f(g(x)) Derivative f'(g(x))g'(x)
What does a cusp look like?
When a function becomes vertical and then virtually doubles back on itself. Such pattern signals the presence of what is known as a vertical cusp.
How do I find an equation of a line tangent to a curve
Find the coordinates of the point, find the slope at the point (by finding the derivative and plugging x in) then insert into point slope
When does a derivative not exist at 'x' (with a graph)?
Corner Cusp Vertical Tangent Discontinuity
Extreme Value theorem
If f is continuous over a closed interval, then f has maximum an minimum values over that interval.
What does a Vertical Tangent look like?
vertical tangent image
Quotient Rule
Function (f/g) Derivative (gf' - fg')/(g^2)
Why can't you draw a tangent line on a corner?
You can't draw a tangent line because the tangent line from the left and the right will be going different directions.
Find the derivative of the square root of f(x)
The derivative of the square root of a function is equal to the derivative of the radical divided by the double of the root.
Finding the vertical asymptote
When the denominator of the function equals 0.
How do you determine the end behavior model of a polynomial function going to positive or negative infinity?
take the variable with the largest exponent and substitute the variable with the limit
Chain Rule
We use chain rule to find the derivative of the composition of two functions. formula : dy/dx f(g(x)) = f'(g(x))*g'(x)
Quotient rule?
(vu'-uv')/v^2
Product rule?
uv'+vu'
How do we handle negative exponents?
They are moved to the bottom of a fraction to make the exponent positive. When finding derivatives, it's easier to solve when you put a factor from the denominator of the fraction to the top with a negative exponent and use the power rule.
Mean value theorem for derivatives
if f(x) is continuous over [a,b] and differentiable over (a,b), then at some point c is between a and b.
When is a function increasing?
When the first derivative/ slope is positive
Product Rule
Function - fg Derivative - f g' + f' g
Types of discontinuity
Removable Discontinuity: when a point on the graph is undefined or does not fit the rest of the graph (there is a hole) Jump Discontinuity: when two one-sided limits exist, but they have different values Infinite Discontinuity:
When are limits nonexistent?
Jump Discontinuities: both one-sided limits exist, but have different values. Infinite Discontinuities: both one-sided limits are infinite. Endpoint Discontinuities: only one of the one-sided limits exists. Mixed: at least one of the one-sided limits does not exist.
What must be true for a limit to exist?
limit from the left = limit from the right
critical points
Is where there is a point in the domain of a function f at which f'=0 or f' does not exist is a critical point of f. *critical points are not always maximum and minimum values.
Limit
A limit is the value that a function or sequence "approaches" as the input or index approaches some value.
What is the derivative of a position function? How do you find where the function is decreasing?
Speed/Velocity. The function is decreasing when y' is negative (below the x-axis)
How do you interpret a velocity graph to determine speed?
Velocity is the first derivative of position. In order to graph speed from velocity then you need to find the derivative of velocity from the graph. In order to do that you need to reflect the negative terms across the x-axis making them positive.
How do you move a term from the denominator to the numerator?
Make the power of the denominator negative than multiply the denominator by the numerator
How do you find the derivative of an inverse function?
If f and g are inverse functions, then f'(x)=1/(g'(f(x))
Derivative of y
dy/dx
What conditions must be to satisfied for the Mean Value Theorem to be valid?
f(x) is continuous in the interval [a, b] and differentiable in the interval (a, b)
Section 2
(4 cards)