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Calculus For Machine Learning - Coggle Diagram
Calculus For Machine Learning
Understanding Linear and Nonlinear Functions
Calculus help us:
Understanf the steepness at various points
FInd the extreme points in a function
Determine the optimal function that best represents a dataset
A linear function is a straight line
If
m
and
b
are constant values where
x
and
y
are variables then the function for a linear function is:
y = mx+b
In a linear function, the
m
value controls
how steep a line is
while the value controls a line's
y
intercept or where the
line crosses the axis
.
One way to think about slope is a rate of change. Put more concretely, slope is how much the
y
axis changes for a specific change in the x axis. id (
x1, y1)
and
(x2,y2)
are coordinates on line, the slope equation is:
m = (y1-y2)/(x1-x2)
When x1 and x2 are equivalent, the slope is undefined because the division of 0 has no meaning
in mathematics.
Nonlinear functions represent curves, and they're output values ( y) are not proportional to their
input values (x).
A line that intersects two points on a curve is known a
secant line.
The slope between any two given points is known as the instantaneous rate of change. For linear functions, the rate of change at any point on the line is the same. For nonlinear function, the
instantaneous rate of change describes the slope of the line that's perpendicular to the nonlinear function at a specific point.
The line that is perpendicular to the nonlinear function at a specific point is known as the tangent line, and only intersects the function at one point.
Understanding Limits
A limit describes the value a function approaches when the input variable to the function approaches a specific value. A function at a specific point may have a limit even though the point is undefined.
The following mathematical notation formalizes the statement "As x2 approaches 3, the slope between x1 and x2 approaches -3" using a limit:
lime x2-> 3 (f(x2)-f(x1)/(x2-x1)=-3
A defined limit can be evaluated by substituting the value into the limit. Whenever the resulting value of a limit is defined at the value input variable approaches, we say that limit is
defined
.
Properties of Limits
• Sum Rule:
lim x→a[f(x) + g(x)] = lim x→a f(x) + lim x→a g(x)
• Difference Rule
: lim x→a[f(x) − g(x)] = lim x→a f(x) − lim x→a g(x)
•
Constant Function Rule
: lim x→a[cf(x)] = c lim x→a f(x)
Finding Extreme Points
Only on md or pdf