Use linear approximations to estimate the quantity. (a) sin(0.1) (b) sin(1.0) (c) sin((9)/(4))

Name: Problem 8, Chapter 3: Calculus: Early Transcendental Functions
Uploaded: 2020-06-10T04:43:28-08:00
Duration: 9 min 36 s
Channel: Grace Muhihu
Description: Use linear approximations to estimate the quantity. (a) sin(0.1) (b) sin(1.0) (c) sin((9)/(4))

Use linear approximations to estimate the quantity. (a)...

Key Concepts

Derivative

The derivative of a function at a point measures the rate at which the function changes with respect to changes in its input. In the context of linear approximation, the derivative provides the slope of the tangent line at the chosen point, which is crucial for constructing the approximation formula f(x) ? f(a) + f'(a)(x – a).

Local Linearity

Local linearity is the property of differentiable functions whereby, in a very small neighborhood around a point, the function behaves almost like a straight line. This characteristic underpins the accuracy of linear approximations over small intervals, as the true function and its tangent line have nearly identical values near the point of tangency.

Linear Approximation

Linear approximation, or linearization, is a method used to estimate the value of a function near a given point by using the function's tangent line at that point. This approach relies on the idea that for values of x close to a known point, the function can be well-approximated by its first-order Taylor expansion about that point.

Transcript

00:01 Okay, so for this question, we're going to use the estimation of sine of zero.

00:10 So we're going to let let x sub -zero be equal to zero.

00:21 So, and the function that we're going to use for this problem is going to be f of x equal to sine x.

00:28 So f of x is going to be equal to sine x.

00:36 Okay, so then from here what we're going to do is we're going to use the linear approximation to try to estimate sign of 0 .1.

00:48 So the formula that we want to use is l of x is approximately equal to f of x, which is equal to f prime of x, which is equal to f prime of x, times x minus x sub 0 plus f of x sub 0.

01:14 Okay, so then from here, what we need to do is we need to find f of x of zero, which is going to be.

01:26 So we plug, so remember f of x is sine x.

01:29 So if we plug zero in, we get sign of zero, which is zero.

01:33 And then the other piece of information we need to find is f prime of zero.

01:38 But first we need to find the derivative.

01:40 So derivative of sine of x is cosine x.

01:45 So now if we plug in zero, so f prime of zero is equal to cosine zero, which will then give us one.

01:55 Now we can go ahead and plug these into our l of x.

01:58 So l of x is going to be equal to, so 1, which is f prime of 0 times x minus 0 plus f of 0 which is also 0.

02:14 So with this we get that l of x is equal to x.

02:21 So that's going to be our first linear approximation.

02:28 Okay for this part.

02:34 So then what we need to find is l of 0 .1.

02:40 So l of 0 .1 is just going to be 0 .1.

02:48 Okay, so that's going to be our approximation for sign of 0 .1.

02:56 So now for the next part, so for the next part, we want sign of 1 .0.

03:11 Okay, so for this, so instead of letting x be equal to 0, we're going to let x be pi over 3 because it's much closer to our desired quantity, which is 1.

03:22 Because pi over 3 is very close to 1, so that's why i'm using that value for x.

03:29 So in this case, we're going to do the same thing...

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