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[M] Repeat Exercise 37 for the functions$\mathbf{f}(t)=3 \sin t-4 \sin ^{3} t$$\mathbf{g}(t)=1-8 \sin ^{2} t+8 \sin ^{4} t$$\mathbf{h}(t)=5 \sin t-20 \sin ^{3} t+16 \sin ^{5} t$in the vector space $\operatorname{Span}\left\{1, \sin t, \sin ^{2} t, \ldots, \sin ^{5} t\right\}$

Observed that most of the function values of the function $1+\mathbf{h}(t)-\sin (5 t)$ are moving veryclosed to the $y=1$ line. Therefore, our conjecture is verified.

Calculus 3

Chapter 4

Vector Spaces

Section 1

Vector Spaces and Subspaces

Vectors

Campbell University

Harvey Mudd College

University of Nottingham

Boston College

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in order to find our double prime unalterable crime. First, we need to find our prime of tea. So to find our priority would just do component lies. Differentiation. So derivative of co sign is negative signs. We're gonna have negative three because of chain role a three science three t and then comma derivative Sinus cosigned. So we're gonna have for co sign of 14 1,000,000 comma And then again, derivative of consign his negative science will have, um negative six sign six t now we confined our double prime, our double prime here. So we're going to, um, it's bye bye. Three again and then derivative coz I or Sinus cosigned, then here. Derivative coziness. Negative science. We're gonna have a negative 16 sign for tea. And then lastly, we're going to have a negative 36. Ah, design of six t here. So that's gonna be our double prime a t. Next. In order to find our triple Prime a T, we're going to take the derivative of that. So here, derivative of co sign is negative sign. So we're going to have a positive 27 sign of three t here, then four times 16 is negative 60 or 64 on derivative of science Because science is gonna be negative. They're negative. 64 co sign of four t and then lastly, six times 36. That is gonna be 216. And derivative clothes line is negative. Sign. We're gonna have negative 216. Sign of six t. So this is going to be our our triple prime here. Sorry. This should be a positive since drew it of, Ah, co sign is negative sign, okay?

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