Related Rates Problems

Lecture 18: Applications of the Derivative

1. Related Rates

• Examines how the rates of change of different variables are related.

Example 1

• A 10 ft ladder rests against a wall, the bottom slides away at 1 ft/s.
• It seeks to determine how fast the top is sliding down when the bottom is 6 ft from the wall.
• Pythagorean Theorem: ( x^2 + y^2 = 10^2 )
• Differentiating with respect to ( t ) yields ( \frac{dy}{dt} = -\frac{x}{y} \frac{dx}{dt} ).
• When ( x = 6 ) ft, ( y = 8 ) ft, and ( \frac{dx}{dt} = 1 ) ft/s, then ( \frac{dy}{dt} = -\frac{3}{4} ) ft/s.

Example 2

• Air is pumped into a spherical balloon at 100 cm³/s. Goal is to find how fast the radius is increasing when the diameter is 50 cm.
• Volume of a sphere: ( V = \frac{4}{3} \pi r^3 )
• Differentiating with respect to ( t ) results in: ( \frac{dr}{dt} = \frac{1}{4 \pi r^2} \frac{dV}{dt} )
• When ( r = 25 ) cm and ( \frac{dV}{dt} = 100 ) cm³/s, then ( \frac{dr}{dt} = \frac{1}{25 \pi} ) cm/s.

2. Linear Approximation and Differentials

Linear Approximation

• ( f(x) \approx f(a) + f'(a) (x-a) ) for ( x ) near ( a ) approximates the function ( f(x) ) using its tangent line at ( x=a ).

Example 3

• The function ( f(x) = \sqrt{x+3} ) is approximated at ( a=1 ) to estimate ( \sqrt{3.98} ) and ( \sqrt{4.05} ).
• The derivative ( f'(x) = \frac{1}{2\sqrt{x+3}} ) gives ( f'(1) = \frac{1}{4} ) and ( f(1) = 2 ).
• Thus, ( \sqrt{x+3} \approx 2 + \frac{1}{4} (x-1) ).
• Approximation of ( \sqrt{3.98} ) yields approximately 1.995.
• Approximation of ( \sqrt{4.05} ) yields approximately 2.0125.

Lecture 1: Course Introduction

• Provides mathematical tools for understanding machine learning (ML) algorithms.
• Teaches how to analyze existing ML algorithms and design new ones.

What the course covers:

• Linear Algebra: Covers matrices, vectors, norms, linear transformations, projections, Eigendecomposition, and SVD.
• Vector Calculus: Includes gradients, Hessians, Jacobians, linearization, Taylor approximations, and constrained optimization.
• Probability and Statistics: Deals with random variables, distributions, Bayesian inference, and hypothesis testing.

Prerequisites

• Calculus: Knowledge of differentiation and integration are required.
• Linear Algebra: Basic operations with vectors and matrices are presumed.
• Programming: Experience is beneficial for programming assignments.

Course Logistics

• Lectures held on Tuesdays and Thursdays from 11:00 am - 12:20 pm in NVIDIA Auditorium.
• The course website is on Canvas.
• Canvas provides lecture videos, slides, and assignments.
• Piazza is used for Q&A.

Instructors:

• Instructor: Qi Lei ([email protected]). Office hours: Thursday 1:00 - 2:00 pm at GDC 3.416
• TA: Jinkun Chen ([email protected]). Office hours: Wednesday 3:30 - 4:30 pm at GDC 1.304
• TA: Hyunsuk Kim ([email protected]). Office hours: Friday 2:00 - 3:00 pm at GDC 1.304

Discussion forum (Piazza):

• Students are encouraged to ask and answer questions.
• Check if the question has already been answered before posting.
• Make questions specific by providing context.

Assignments:

• Homework (45% of grade): Assigned bi-weekly; combines mathematical derivations and programming.
• Discussing homework is permitted but solutions and code must be original.
• Late homework incurs a 20% penalty per day, up to 3 days; no credit after.
• Midterm Exam (25% of grade): Held in class on Oct 17th (provisional).
• Final Exam (30% of grade): In class, during final exam week.

Textbook

• No required textbook; recommended books include:
• "Linear Algebra and Its Applications" by Gilbert Strang
• "Mathematics for Machine Learning" by Deisenroth, Faisal, Ong (available online)
• "Pattern Recognition and Machine Learning" by Christopher Bishop (freely available online with UT EID)
• "The Elements of Statistical Learning" by Hastie, Tibshirani, Friedman (freely available online) Relevant reading materials will be posted on Canvas.