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# On page 431 of $Physics: Calculus,$ 2d ed., by Eugene Hecht (Pacific Grove, CA: Brooks/ Cole, 2000), in the course of deriving the formula $T = 2\pi \sqrt {L/g}$ for the period of a pendulum of length $L,$ the author obtains the equation $a_T = -g \sin \theta$ for the tangential acceleration of the bob of the pendulum. He then says, "for small angles, the value of $\theta$ in radians is very nearly the value of $\sin \theta;$ they differ by less than $2\%$ out to about $20^o."$(a) Verify the linear approximation at 0 for the sine function:$\sin x \approx x$(b) Use a graphing device to determine the value of $x$ for which $\sin x$ and $x$ differ by converting from radians to degrees.

## (a) $L(x)=x$(b)

Derivatives

Differentiation

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##### Catherine R.

Missouri State University

##### Heather Z.

Oregon State University

##### Samuel H.

University of Nottingham

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### Video Transcript

okay, part ay. So the first thing we know is that we can use the Formula 000 plus f prime of zeros one times X minus zero, which gives us acts. So Olive Axe is simply equivalent to X copping for my graphing calculator. Now we know we have X minus Sign axe over. Sign X is 0.2 So to figure out where this equals again using a graphing calculator, we get closer minus your 0.344 times 1 80 over pie gives us 19.7 degrees. So, in other words, remember, this is between negative 0.3440 point 344 because it's plus or minus 3440.344

#### Topics

Derivatives

Differentiation

##### Catherine R.

Missouri State University

##### Heather Z.

Oregon State University

##### Samuel H.

University of Nottingham

Lectures

Join Bootcamp