Univariate Calculus: Derivatives and Linear Approximations

Name: Univariate Calculus: Derivatives and Linear Approximations
Uploaded: 2025-01-30T11:07:27.000Z
Duration: 40 min 51 s

Machine Learning Foundations: Understanding Derivatives and Linear Approximations

Introduction to Derivatives

The lecture focuses on univariate calculus, specifically the concept of derivatives and their connection to linear approximations.

A differentiable function f: mathbbR to mathbbR is defined, with the derivative expressed as:

[

f'(x^) = lim_x to x^ f(x) - f(x^)/x - x^

]

Understanding Linear Approximations

When x approx x^* , we can simplify the limit expression, leading to an approximation of f(x) .

The alternative expression for linear approximation is given by:

[

f(x) approx f(x^) + f'(x^)(x - x^*)

]

Key Concepts in Linear Approximation

This approximation holds true when x is close to x^* , allowing us to treat constants like f(x^) and f'(x^).

The notation l_x^(f)(x) = f(x^) + f'(x^)(x - x^) represents a function that approximates f(x).

Graphical Illustration of Linear Approximation

Using the function f(x)=x^2 , with a chosen point at x^*=1:

The derivative at this point is calculated as f'(1)=2.

Calculating the Linear Approximation

The linear approximation formula becomes:

l_1(f)(x)=1+2(x−1)

Simplifying gives us:

l_1(f)(x)=2x−1

Validity of the Approximation

At points near x^*=1, the red line (linear approximation graphically represented by l_1(f)(x)) closely follows the black curve ( y=x^2).

As you move further from this point, accuracy decreases; for example, at x = 2, there’s a noticeable gap between actual values and approximated values.

Characteristics of Linear Functions

The linear function derived from our approximation has a constant slope due to its form being based on derivatives.

This relationship illustrates why it’s termed a "linear" approximation—its structure reflects that of a straight line.

Conclusion on Tangents vs. Linear Approximations

Understanding Tangent Lines and Linear Approximations

Definition of Tangent Lines

A line is defined as a tangent to a set if it touches the first center; formal definitions are avoided for simplicity.

The red line represents the graph of the linear approximation of function f around point x^* , indicating it is tangent to the graph of f .

Key Concepts in Linear Approximations

The point where these two sets touch is denoted as (x^, f(x^)) , which defines the tangent line and its relationship with linear approximations.

Understanding linear approximations involves recognizing how they form tangent lines to graphs, providing foundational knowledge for calculus applications.

Examples of Linear Approximations

Sine Function Approximation

For f(x) = sin x around x^* = 0 :

Derivative: f'(x) = cos x ; thus, at x^* = 0, f'(0) = 1.

Therefore, the approximation becomes:

f(x) approx f(0) + f'(0)(x - 0) = 0 + 1(x - 0) = x.

Valid when x approx 0. This leads to the famous limit result:

lim_theta to 0 sin(theta)/theta = 1.

Exponential Function Approximation

For f(x) = e^x around x^* = 0:

The approximation yields:

e^x approx e^0 + (x - 0)(f'(0)) = 1 + (1)(x).

This holds true only near zero, emphasizing that approximations are context-dependent.

Logarithmic Function Approximation

For logarithm function:

Letting f(x) = log(1+x):

Derivative: f'(x)=1/1+x; hence at point zero, it simplifies to:

Logarithmic approximation gives us:

log(1+x)approx x, valid around zero.

Polynomial Function Approximation

Considering polynomial functions like:

For general case off(x)= (1+x)^r:

At point zero, we find:

The derivative leads to an approximation formula:

Thus, we conclude that this also holds true near zero.

– Resulting in:

– (1+x)^r ≈ 1 + r*x.

– Validity remains close to zero.

Application of Linear Approximations in Problem Solving

Exercise Example on Power Calculation

An exercise asks for an approximate value for (0.99)^7:

– Recognizing this can be expressed as a linear approximation using:

– Setting up as (1 + (-0.01))^7.

– Applying linear approximation results in estimating it as approximately equal to:

– ≈ .93.

– This provides insight into practical applications such as financial decay over time through compound interest simplifications.

Conclusion on Practical Implications

The discussion emphasizes how linear approximations simplify complex calculations while maintaining accuracy within small ranges.

Understanding Derivatives and Linear Approximations

The Validity of Percentage Loss in Approximations

A small percentage loss, such as one percent, can be managed effectively within valid approximations. However, larger losses (e.g., ten percent) may invalidate these approximations due to the underlying mathematical principles.

Limitations of Linear Approximations

The linear approximation l_x^(f) is only accurate near the point x = x^ . As you move away from this point, the evaluation of l_x^*(f) diverges from the actual function value f(x) , indicating a limitation in its applicability for broader ranges.