Homework01_Soln_Math241_Spring10

# Homework01_Soln_Math241_Spring10 - Homework 1 – A Brief...

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Homework 1 – A Brief Calculus I and II review Due Thursday, January 21st 1. Chain Rule (a) Let h ( t ) := sin(cos(tan t )). Find the derivative with respect to t Solution : d dt ( h ( t )) = d dt (sin(cos(tan t ))) = cos(cos(tan t )) d dt (cos(tan t )) = cos(cos(tan t ))(- sin(tan t )) d dt (tan t ) = cos(cos(tan t ))(- sin(tan t ))sec 2 t (b) Let s ( x ) := 4 √ x where x ( t ) := ln( f ( t )) and f ( t ) is a differentiable function. Find ds dt . Solution : Using Leibniz notation, ds dt = ds dx dx dt . So, ds dt = 1 4 x 3 / 4 · f ( t ) f ( t ) . But we need make sure ds dt is a single variable function of f . So, ds dt = 1 4[ln( f ( t ))] 3 / 4 · f ( t ) f ( t ) . 2. Use implicit differentiation to find dy dx when sin( x + y ) = y 2 cos x. Solution : Recall that we treat y as a function of x . I.e. y ( x ). sin( x + y ) = y 2 cos x d dx (sin( x + y ) = y 2 cos x ) d dx (sin( x + y ( x ))) = d dx ( y 2 ( x )cos x ) cos( x + y ) d dx ( x + y ( x )) = 2 y dy dx cos x + y 2 (- sin x ) cos( x + y ) 1 + dy dx ¶ = 2 y cos x dy dx- y 2 sin x ) cos( x + y ) + cos( x + y ) dy dx = 2 y cos x dy dx- y 2 sin x cos( x + y ) dy dx- 2 y cos x dy dx =- cos( x + y )- y 2 sin x dy dx =- cos( x + y )- y 2 sin x ) cos( x + y )- 2 y cos x 3. Parametrized curves (a) Describe the curve given parametrically by ( x := 5sin(3 t ) y := 3cos(3 t ) , when 0 ≤ t < 2 π 3 . What happens if we allow t to vary between 0 and 2 π ? Solution : Note that ‡ x 5 · 2 + ‡ y 3 · 2 = sin 2 (3 t ) + cos 2 (3 t ) = 1. Hence, this parametrizes (at least part of) the ellipse ‡ x 5 · 2 + ‡ y 3 · 2 = 1. By examining differing values of t in 0 ≤ t ≤ 2 π 3 , we see that the parametrization above travels the ellipse in a clockwise fashion exactly once....
View Full Document

## This note was uploaded on 02/21/2011 for the course MATH 241 taught by Professor Kim during the Spring '08 term at University of Illinois, Urbana Champaign.

### Page1 / 6

Homework01_Soln_Math241_Spring10 - Homework 1 – A Brief...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online