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Given 7(t) = (sin³ t, cos³ t, sin² t), 0 < t s (HINT: When finding ||r(t)||, note the common factor cos² t sin? t inside the square root, which is positive on 0 sts", i.e. Vcos² t sin² t = cos t sin t) a. Find the vectors T, N and B when t = " b. Find the curvature when t =

Given 7(t) = (sin³ t, cos³ t, sin² t), 0 < t s (HINT: When finding ||r(t)||, note the common factor cos² t sin? t inside the square root, which is positive on 0 sts", i.e. Vcos² t sin² t = cos t sin t) a. Find the vectors T, N and B when t = " b. Find the curvature when t =

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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**Problem Statement:** Given the vector function \( \mathbf{r}(t) = (\sin^3 t, \cos^3 t, \sin^2 t) \) for the interval \( 0 \leq t \leq \frac{\pi}{2} \). **Hint:** When finding \( \|\mathbf{r}'(t)\| \), note the common factor \( \cos^2 t \sin^2 t \) inside the square root, which is positive on \( 0 \leq t \leq \frac{\pi}{2} \), i.e., \( \sqrt{\cos^2 t \sin^2 t} = \cos t \sin t \). **Questions:** a. Find the vectors \(\mathbf{T}\), \(\mathbf{N}\), and \(\mathbf{B}\) when \( t = \frac{\pi}{4} \). b. Find the curvature when \( t = \frac{\pi}{4} \).
Transcribed Image Text:**Problem Statement:** Given the vector function \( \mathbf{r}(t) = (\sin^3 t, \cos^3 t, \sin^2 t) \) for the interval \( 0 \leq t \leq \frac{\pi}{2} \). **Hint:** When finding \( \|\mathbf{r}'(t)\| \), note the common factor \( \cos^2 t \sin^2 t \) inside the square root, which is positive on \( 0 \leq t \leq \frac{\pi}{2} \), i.e., \( \sqrt{\cos^2 t \sin^2 t} = \cos t \sin t \). **Questions:** a. Find the vectors \(\mathbf{T}\), \(\mathbf{N}\), and \(\mathbf{B}\) when \( t = \frac{\pi}{4} \). b. Find the curvature when \( t = \frac{\pi}{4} \).
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