3.1-3.5 Unit 3

Question 1

Definition of derivative at a point

Answer 1

f ‘(c) = lim x->c (f(x)-f(c)/x-c)

Question 2

a. point slope form
b. and what y1, m, and x1 are
c. when do you use this

Answer 2

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

Question 3

a. forward difference quotient
b. backward difference quotient
c. symmetric difference quotient

Answer 3

a. deltay/h
b. deltay/h
c. deltay/2h

Question 4

Calculator stuff involved in finding f’(2) of 2x^2 + 5x - 2

Answer 4

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

Question 5

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

Answer 5

a. math
b. x y1 and x
c. x y1 and 1 (or whatever number your trying to find)

Question 6

Obligatory cry in despair about graphs time

Answer 6

they’re difficult, its sad

Question 7

Derivative of a function (equation)

Answer 7

f’(c) = lim h->0 ((f(x+h) - f(x)) / h)

Question 8

a. What equation do you use to find f’(x) ?
b. What equation do you use to find f’(c)?

Answer 8

a. derivative at a function
b. derivative at a point

Question 9

If asked to use a definition of derivative do you have to do it the long way??

Answer 9

YES

Question 10

(a+b)^2

Answer 10

a^2 +2ab+b^2

Question 11

f(x) = square root of x
make it nicer

Answer 11

f(x) = x^1/2

Question 12

Derivative of the power function in theory and in ex

Answer 12

if f(x) = x^n then f’(x) = nx^n-1
in ex: f(x) = x^7 -> f’(x) =7x^6

Question 13

a. Velocity
b. Units
c. How does x’(t) relate to velocity?

Answer 13

a. how quickly something changes position
b. distance/time
c. it is velocity (first derivative)

Question 14

Speed

Answer 14

|v| has not direction

Question 15

a. Acceleration
b. Units
c. How does x’‘(t) relate to acceleration?

Answer 15

a. how quickly something changes velocity
b. distance/time^2
c. it is acceleration (derivative of velocity v’(t))

Question 16

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

Answer 16

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

Question 17

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

Answer 17

a. f’(c or x)
b. that they have the same or different signs

Question 18

a. for function y what are two ways to say the first derivative?
b. second derivative?

Answer 18

a. dy/dx and (d/dx)y
b. d^2y/dx^2 and (d^2/dx^2)y

Question 19

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

Answer 19

it is missing +C on the end

Question 20

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 ___

Answer 20

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

Question 21

What is d/dx(15-x^2) asking you to do?

Answer 21

find the derivative of 15-x^2

Question 22

Find the derivative of:
a. y = sin2pi/3
b. e^13
c. e^13x

Answer 22

a. 0
b. 0
c. e^13