3.1-3.5 Unit 3 Flashcards
Definition of derivative at a point
f ‘(c) = lim x->c (f(x)-f(c)/x-c)
a. point slope form
b. and what y1, m, and x1 are
c. when do you use this
a. y - y1 = m(x-x1)
b. y1 is f(c) x1 is c and m is what you get in the definition of derivative at a point equation
c. when asked to find the equation of the tangent line
a. forward difference quotient
b. backward difference quotient
c. symmetric difference quotient
a. deltay/h
b. deltay/h
c. deltay/2h
Calculator stuff involved in finding f’(2) of 2x^2 + 5x - 2
1st put y1 as 2x^2 + 5x - 2
2nd hit calc button and choose dy/dx
3rd choose point x=2 (just type it in immediately you don’t need to use trace)
4th it gives you the answer, which is 13
a. nDeriv is where in the calculator?
b. what do you put in the open spots in the general case?
c. in the specific? use f’(1) for example
a. math
b. x y1 and x
c. x y1 and 1 (or whatever number your trying to find)
Obligatory cry in despair about graphs time
they’re difficult, its sad
Derivative of a function (equation)
f’(c) = lim h->0 ((f(x+h) - f(x)) / h)
a. What equation do you use to find f’(x) ?
b. What equation do you use to find f’(c)?
a. derivative at a function
b. derivative at a point
If asked to use a definition of derivative do you have to do it the long way??
YES
(a+b)^2
a^2 +2ab+b^2
f(x) = square root of x
make it nicer
f(x) = x^1/2
Derivative of the power function in theory and in ex
if f(x) = x^n then f’(x) = nx^n-1
in ex: f(x) = x^7 -> f’(x) =7x^6
a. Velocity
b. Units
c. How does x’(t) relate to velocity?
a. how quickly something changes position
b. distance/time
c. it is velocity (first derivative)
Speed
|v| has not direction
a. Acceleration
b. Units
c. How does x’‘(t) relate to acceleration?
a. how quickly something changes velocity
b. distance/time^2
c. it is acceleration (derivative of velocity v’(t))
a. How do you tell if an object is increasing or decreasing?
b. How do you tell if the object is speeding up or slowing down
a. if v is positive its increasing, if v is negative its decreasing
b. if the signs (positive or negative) are the same for acceleration and velocity then the object is speeding up. If the signs are not the same then the object is slowing down
a. What do you have to keep in your answer and work for derivative at a function and at a point?
b. What do you have to have in your answer about if an object is speeding up or slowing down
a. f’(c or x)
b. that they have the same or different signs
a. for function y what are two ways to say the first derivative?
b. second derivative?
a. dy/dx and (d/dx)y
b. d^2y/dx^2 and (d^2/dx^2)y
If asked a question like:
f’(t) = 4t^3 - 3t^2 + 2t - 1
what could f(x) = ?
what is this answer missing?:
t^4 - t^3 + t^2 - t
it is missing +C on the end
Graphing derivative stuff:
a. For a diagonal line (ex. y=x) and a horizontal line (ex. y=2) what does the derivative graph look like?
b. If the graph looks parabolic (x^2) then the derivative graph is
c. derivative graph is _____ on high and low points
d. If the slope of the function is steep the derivative graph will be ___
e. If the slope of the function is positive the derivative graph will be _____, if the slope of the function is negative the derivative graph will be ___
a. derivative graph is a horizontal line for both, it goes where the slope is for the function (ex. y=x has a derivative graph of a horizontal line on y=1 b/c the slope of y = x is one, and y=2 has a line on 0 b/c the slope is 0)
b. diagonal line, if the graph is y = x^2, with the slope 2x
c. 0
d. steep
e. on the top of the graph, on the bottom of the graph
What is d/dx(15-x^2) asking you to do?
find the derivative of 15-x^2
Find the derivative of:
a. y = sin2pi/3
b. e^13
c. e^13x
a. 0
b. 0
c. e^13
