MFDS II

Question 1

Continuity

Answer 1

A function f(x) is said to be continuous at a point x=a if the following conditions are met:
f(a) is defined
lim(x→a) f(x) exists
lim(x→a) f(x) = f(a)

Question 2

Differentiability

Answer 2

A function f(x) is said to be differentiable at a point x=a if the derivative f’(a) exists. The derivative is defined as:
f’(a) = lim(h→0) [f(a+h) - f(a)]/h

Question 3

Continuity and differentiability relation

Answer 3

If a function is differentiable at a point, it is also continuous at that point.
If a function is continuous at a point, it is not necessarily differentiable at that point

Question 4

Real analysis applications

Answer 4

1 Machine learning
Linear regression (OLS estimator)
Gradient descent (opti loss
function)
2 Time series analysis
Fourier series (decom. time ser
data)
3 Data compression
PCA algorithm
SVD algorithm
4 Signal processing
Frequency noise reduction by
filtering

Question 5

Convergence

Answer 5

Limit of nth term of a series exists and is finite

Question 6

monotonic sequence

Answer 6

A monotonic sequence is a sequence whose terms are either consistently increasing or consistently decreasing

Question 7

gradient

Answer 7

vector that points to maximum rate of increase of function
del f(x,y) = (df/fx, df/dy)

Question 8

divergence

Answer 8

measures the extetent to which vector field spreads out from a point. it is scalar quanityt and represents rate of change of field’s density

Question 9

curl

Answer 9

measures the rotation or circulation at a point. It results in another vector that represents axis and magnitude

Question 10

directional derivative is a scalar quantity

Answer 10

The directional derivative of a function is a scalar quantity that represents the rate of change of the function in a specific direction.
example:
phi = x2yz3 at point 1,1,1
v = i + 2j - 3k

Question 11

Applications of partial derivatives in data science

Answer 11

Gradient descent (minimize loss function)
Neural Network Optimization(gradient of losss function wrt network weights)
Regression analysis (best fit data by minimizing error function)
Support vector machine (optimize hyperparameters for SVM)
Expectation Maximization Algorithm (iteratively update parameter estimate)