Lecture 21

Question 1

how do you express the annual growth rate of a function as a differentiable equation if the growth rate is 1.2%?

Answer 1

1. divide the rate by 100 to get it in decimal form
2. make the differential equation using Leibniz form
   ex. dP/dt=0.012P

Question 2

what does it mean if a word problem uses the word difference?

Answer 2

the subtraction between two variables

Question 3

what is the variable for the constant of proportionality?

Answer 3

k

Question 4

what function notation should you use if you are trying to find the differential equation from a word problem?

Answer 4

Leibniz notation

Question 5

if you are given an autonomous differentiable equation and Y(t)=20, how do you verify that 20 is a solution?

Answer 5

1. plug 20 into the differentiable equation
2. if the answer is 0, then 20 is a solution

Question 6

what does it mean if a number is an equilibrium solution in terms of rate?

Answer 6

when the rate of change of y and x are constant

Question 7

what does stable equilibrium mean?

Answer 7

when a system tends to return to its original position after being disturbed

Question 8

what does it mean if a function is autonomous?

Answer 8

if the right side of the function only has a y variable

Question 9

how do you find all the solutions for an autonomous equation?

Answer 9

find all the y values from the derivative

Question 10

what is unstable equilibrium?

Answer 10

when a disturbance causes a significant departure from the equilibrium position

Question 11

how do you create a slope field?

Answer 11

1. create a table of values with two columns, one with (x,y) and the other with y’
2. use the derivative function and the points made up to find the slope at each point
3. draw small line segments on a graph to represent the slope at each point

Question 12

how can you tell that a graph matches a differentiable equation?

Answer 12

estimate slopes at various parts of the graph and see if they match

Question 13

what does it mean if there is an empty row on a slope field?

Answer 13

there is a vertical tangent at the y value where the empty row exists

Question 14

How do you use the Euler method?

Answer 14

1. create a table with three columns, one with x, one with y and one with dy/dx*change in x plus y
2. put the initial point in the table right away
3. use what is given to continue the table until you reach your point of interest

Question 15

how do you determine whether Euler’s method will give an over or under-estimate of an autonomous function?

Answer 15

1. plug the point of interest in the derivative
2. if the slope is increasing then it is an underestimate
3. if the slope is decreasing, then it is an overestimate

Question 16

what is the linear approximation formula?

Answer 16

y2-y1=m (x2-x1)

Question 17

what does it look like in a function if the rate of change of one variable is inversely proportional to another variable?

Answer 17

when one variable is in the numerator and the other is in the denomination