Section 1
Preview this deck
Power Rule
Front
| 5 | 0 | ||
| 4 | 0 | ||
| 3 | 0 | ||
| 2 | 0 | ||
| 1 | 0 |
Active users
0
All-time users
0
Favorites
0
Last updated
6 years ago
Date created
Mar 1, 2020
Cards (87)
Section 1
(50 cards)
Power Rule
Function - x^n Derivative - 〖nx〗^(n-1)
Rules of Piecewise Functions
A Piecewise function is made up of sub-functions that apply to a certain interval of the main function's domain. First look at the conditions on the right to see where x is. Then just plug that number into the equation.
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]
Implicit differentitation
(image)
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.
Power Rule
If function f is f(x) = x^n, where n is any integer, then f' (x) = n·x^n-1.
Power rule
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.
how to find identify the original function of a graph
How do we handle negative exponents?
Negative exponents 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.
If the definition of the derivative is recognized, what does it guarantee?
Unit Circle
Since C = 2πr, the circumference of a unit circle is 2π. A unit circle is a circle with a radius of one. Frequently, especially in trigonometry, the unit circle is the circle of radius one centered at the origin (0, 0) in the Cartesian coordinate system.
Quotient Rule
Function (f/g) Derivative
Derivative of e^x
e^x * derivative of argument
Chain Rule (Using d/dx)
Product rule?
uv'+vu'
Chain Rule (Using ' )
Function f(g(x)) Derivative f'(g(x))g'(x)
What does a Corner look like?
Corner at (17, 600) and (18, 530)
Power rule
X^2 --> 2X^2-1 ---> 2X
Product Rule
uv' + vu'
Rate of Change
Expressed as a ratio between a change in one variable relative to a corresponding change in another. When the function y=F(x) is concave up, the graph of its derivative y=f'(x) is increasing. When the function y=F(x) is concave down, the graph of its derivative y=f'(x) is decreasing.
Second Derivative
Take first derivative. Then, find the derivative of the first derivative. f'(x), then f''(x).
What does a jump discontinuity look like?
Quotient rule?
(vu'-uv')/v^2
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. x = 0, -2, or These three x-values are the critical numbers of f.
How do I find an equation of a line tangent to a curve
1.) Calculate the slope of a secant through P and a point Q nearby on the curve. 2.)Find the limiting value of a secant slope (if it exists) as Q approaches P along the curve. 3.)Define the slope of the curve at P to be this number and define the tangent to the curve at P to be the line through P with this slope.
Product Rule
Function - fg Derivative - f g' + f' g
How to find derivative?
Solve a problem for (dy/dx). Solutions may involve power, product, chain or quotient rule.
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)
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.
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
Derivative of tangent inverse
When does a derivative not exist at 'x' (with a graph)?
Corner Cusp Vertical Tangent Discontinuity
Derivative rules
Limit
A limit is the value that a function or sequence "approaches" as the input or index approaches some value.
The derivative of the function f at the point x=a is the limit... if the limit exists.
picture on email
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.
Definition of a limit
definition of a limit
What are the derivatives of trig functions?
sin(x) = cos (x); cos (x) = -sin(x); tan(x) = sec^2(x)
Reciprocal Rule
Function 1/f Derivative −f'/f2
Chain Rule
(F o G)' (x) = f'(g(x)) *g' (x)
Recognizing Implicit Differentiation
Used when an equation contains a variable besides x For example: 3x + 4y = 12 derivative of y= dy/dx
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.
Difference Rule
Function - f - g Derivative - f' − g'
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)
What does a Vertical Tangent look like?
vertical tangent image
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
Sum Rule
Function - f + g Derivative - f' + g'
What are discontinuities? When are limits nonexistent?
Limits dont exist when the values from the left and righ are3 no equal
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:
Section 2
(37 cards)