Unit 3 - Differentiation: Definition and Basic Derivative Rules Flashcards
Equation of a tangent line to a graph at a point
f’(c)(x-c)+f(c)
How do you find the normal?
Find the slope of the tangent, and take the negative reciprocal.
What is the normal when the tangent line has a slope of 0?
The normal is a vertical line, represented as x=c
What is the equation for the average rate of change?
average rate of change = (f(x)-f(c)) / (x-c) where x=/=c
What letter represents distance in Calculus?
s
What is average velocity?
Change in position with respect to time.
Difference between speed and velocity?
Speed is absolute. Velocity is signed (can be positive or negative)
What is instantaneous velocity?
The limit of average velocity as Delta t approaches 0.
How do you find the derivative when given a table of data?
Use the two closest independent values, and find the secant. (this approximates the tangent)
How do continuity and differentiability relate?
If a function is differentiable at c, it is continuous as c (f must be continuous at c for f to be differentiable at c. If a function is not continuous at c, it is not differentiable at c. If a function is continuous at c, it is not necessarily differentiable.
Differentiability implies continuity, but continuity does not imply differentiability.
What is the derivative of e^x
e^x
Simple power rule
x^n = nx^(n-1) Does not work for exponentials (n^x).
Sum/Difference rule.
f(x)=g(x)+-h(x). If g(x) and h(x) are differentiable, then f(x) is differentiable and f’(x)=g’(x)+-h’(x)
Notations for derivatives.
f’(x)
y’
dy/dx
(d/dx)y
(d/dx)f(x)
Df(x)
Derivative of a constant.
If f is a constant function f(x)=a, then f’(x)=0
What is the derivative of the natural logarithm?
y=ln x, y’=1/x
What is the product rule?
(uv)’=uv’+u’v
What is the quotient law?
(u/v)’=(vu’-uv’)/(v^2)
Higher order derivatives
y^(n)
f^(n) (x)
d^(n) y / dx^(n)
d / dx^(n)
derivative of sin(x)
cos(x) over all real numbers
derivative of cos(x)
-sin(x) over all real numbers
derivative of tan(x)
sec^2 (x) where x does not equal (2k+1)/(2) * Pi, k an intiger
derivative of cot(x)
-csc^2 (x) where x does not equal nPi, n an intiger
derivative of csc(x)
-csc(x) * cot(x), where x does not equal nPi, n an intiger
derivative of sec(x)
sec(x) * tan(x), where x does not equal (2k+1)/(2) * Pi, k an intiger
Useful note for remembering some derivatives of trigonometric functions?
If the trigonometric function starts with a c, its derivative has a negative sign
How do you find the derivative using its definition
find the limit as h approaches 0 of (f(x+h)-f(x)) / h
What does it mean of f’(a) (f prime of x at a) does not exist?
The tangent might not exist, might be a vertical line, or might not be unique
What does limit as x approaches c of (f(x)-f(c)/(f-x) equal?
The tangent line to the graph of f at a c, provided the limit exists
Average rate of change equation
(f(x)-f(c))/(x-c), x =/= c
What is average rate of change?
behavior over an interval
What is instantaneous rate of change?
behavior at a number (not over a range)
Average velocity equation
Delta s/ Delta t = (f(t1))-f(t0)) / (t1-t0)
Instantaneous velocity equation
v = limit as delta t approaches 0 of Delta s/ Delta t = limit as t0 approaches t0 of (f(t1))-f(t0)) / (t1-t0)
How to use the definition for derivative of a function at a number?
f’(c) = limit as x approaches c of ((f(x)-f(c))/(x-c)
How to use the definition for a derivative function f’ of a function f?
f’(x) = limit as h approaches 0 of ((f(x+h)-f(x))/h
What does it mean when asked to use “the definition” to find the answer?
Longer form. No shortcuts.
What are the 3 ways a function cannot be differentiable at a point on a continuous function?
The limit from the left and right exist but are not equal (corner). The one-sided limits are both infinite (+ or -) (vertical tangent). The one-sided limits are infinite but one is positive and the other is negative (cusp).
How do you know where the horizontal tangent lines of a graph of f are?
Where f’ is 0.
What are the common errors you make?
Incorrectly reducing exponents. Forgetting negatives/adding them where they don’t exist. Improperly formatting quotient rules. READ THE QUESTION AND KNOW WHAT IT WANTS. Remember that some limits of trigonometric equations involve squares.
Memory trick for quotient rules of derivatives.
Low d-high minus high d-low, draw a line and square below
What do you need to remember about solving for x of a trigonometric equation?
Remember the domain. If none is given, give the general solution.
How would you format a falling object’s speed?
Falling at abs(velocity). Falling refers to the direction, so signs are not needed.
Position… Velocity… what comes next?
Position… Velocity… then acceleration, jerk and snap.
Form 1 and Form 2
todo
What should you watch out for when a question asks for a limit?
Check if it is using the definition of a derivative
For the current unit, do vertical tangents exist
No.
What does m (subscript tangent) represent?
Slope of the tangent. Equal to form 1 of the definition of a derivative
What do you need to remember about graphs?
Check the scale of the axi.
When asked to match a function’s graph to its derivatives graph, what do you do?
Remember that you don’t know the equations. Describe the behaviour of the graphs over intervals and points.
Finding limit of a piece wise function at a point
Use form 1 (x -> c) from the left and right sided limits. Check for continuity as well
Do cusps have derivatives?
No. Derivatives cannot be infinite (for now at least)
How to determine continuity using differentiability.
Wherever f is differentiable, f is continuous. Where f is not differentiable, you need more information to determine continuity.
Difference between at a point and value.
Point are like (x,y). Values are x=… units
