Terms
Slope Fields
Euler's Method
Separation of Variables
Exponential Growth and Decay
100
The rate of change of a function.
What is a derivative
100
What is a slope field
Slope fields are graphical representations of differential equations, allowing visualization of the family of solutions without solving the equations algebraically.
100
What is Euler's Method?
Euler's Method is a numerical technique for approximating solutions to differential equations, particularly when explicit solutions are difficult to find.
100
What is it?
A differential equation is separable if it can be expressed in the form dy/dx = f(x)g(y).
100
What is it?
The rate of change of a quantity y is modeled by dy/dt = ky, where k is a constant.
200
Contains arbitrary constants representing a family of solutions.
What is General Solution
200
dy/dx = y shows what?
the slope at any point is equal to the y-coordinate, which can be represented graphically by drawing line segments at various points on the plane.
200
What does the method rely on?
The method relies on the concept of local linearity, where a differentiable function can be approximated by its tangent line at a given point.
200
What does it involve?
The solution involves rearranging the equation to isolate y and x terms on opposite sides: dy/g(y) = f(x)dx.
200
In 2020 beginning, the fish population was 26244 which has declined by 10% per year. What was the fish population in 2017 beginning?
36000
300
Graphical representation of differential equations' slopes.
what is a Slope Field
300
Slope fields help with?
Slope fields help in understanding the behavior of solutions and can guide the sketching of particular solutions when initial conditions are provided.
300
What can we estimate?
By using the slope from the differential equation, we can estimate the function's value at subsequent points, creating a stepwise approximation of the solution.
300
A pizza is removed from the oven after baking thoroughly, and the temperature of the oven is 350°F. The temperature of the kitchen is 75°F, and after 55 minutes the temperature of the pizza is 340°F. We would like to wait until the temperature of the pizza reaches 300°F before cutting and serving it How much longer will we have to wait?
23.5 minutes
300
A radioactive isotope decays at the rate indicated by the exponential function 
, where 't
' is the time in years and A(t) is the amount of the isotope, in grams, remaining. How long will it take for the isotope to be reduced to half of its original amount?
1500 years
400
A function of the form y = ae^(bx).
what is a Exponential Function
400
How do you sketch a slope field
To sketch a slope field for a given differential equation, calculate the slope at various points and draw small line segments representing these slopes.
400
Given the differential equation dy/dx +x+2 and y(0)=3 Find an approximation for y(1) using Euler's Method
5.25
400
dy/3y+2 = (x^2 - 4) dx
solve for y
y= - 2/3
400
If each term in a geometric sequence is defined as an = -6 x (-1)n-1, what is the sum of the first 20 terms of the sequence?
0
500
Numerical technique for approximating solutions of ODEs.
what is Euler's Method
500
thermoregulation mechanisms
What is internal body temperature
500
Given the differential Equation dy/dx 1/x+2 and y(0)=1. Find an approximation of y(1) using Euler's Method.
1.45
500
y'= 2xy + 3y - 4x- 6
y= 2 + Ce^x2+3x
500
If each term in a geometric sequence is defined as an=3 x (4)n-1, what's the sum of the first ten terms of the sequence?
1048575