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M6 Mastery
M6 Mastery
-Melvin Orichi Socana
Constructing Slope Fields
Slope Fields
- Slope field help us visualize how the function changes across the graph as well as illustrates the slope at every point.
- When constructing a slope field, always use the equation of the derivative to find the slope at each point.
- Make sure to remember when drawing slope lines to check if the slope is positive or negative.
- Top Left -> Bottom Right = Negative
- Bottom Left -> Top Right = Positive
- Horizontal Line = 0 Slope
Question
Question
- Given y = x^2 + 3, construct a slope field from x=-1 to x=2 and y=0 to y=2
Remember the find the derivative of the function to begin constructing the slope field.
If you are having trouble figuring out how to evaluate for the slope, try constructing a table.
Solution
y= x^2 + 3
y' = 2x
This slope field will be based entirely on the value of x.
2
1
-1 1 2
f'(1) = 2 f'(-1) = -2
f'(0) = 0 f'(1) = 2
Seperable Differential Equations
- When solving differential equations, it can sometimes be imperative to use a special method of solving and that is to separate the variables, integrate both sides, and solve for y.
- STEP 1: Seperate the Variables in terms of X and Y
- STEP 2: Integrate both sides of the equation
- When integrating, only put + C at the end of the equation to the right.
- STEP 3: Solve for y to sucessfully complete the question
Seperable DEs
Question
Question
dy/dx = 2(y^2)(x^2)
Solve for f'(x).
First separate the variables in terms of y and x
Integrate both sides of the equation using basic rules of integration
Solve for y using basic algebra
Solution
dy/dx = 2(y^2)(x^2)
Step 1: Separate the variables in terms of x and y
dy/y^2 = *2x^2)(dx)
Step 2: Integrate both sides
int(dy/y^2) = int((2x^2)(dx))
: -1/y = ((2x^3)/3) + C (Only put +C on the right)
Step 3: Solve for y
-1 = (((2x^3)/3) + C)*y
y = -1/((2x^3)/3)+C