Use a calculator in radian mode to complete the following table. What can you conclude about the

Use a calculator in radian mode to complete the following...

Key Concepts

Limit

A limit describes the behavior of a function as its input approaches a particular value. In calculus, analyzing limits helps determine the value that a function approaches, even if the function is not defined at that point. This concept is fundamental in the study of continuity and the derivation of derivatives.

Small Angle Approximation

Near zero, trigonometric functions can be approximated by simpler expressions. For instance, cosine can be approximated by 1 ? (?²/2) when ? is small (and measured in radians), which is key to understanding how functions involving cosine behave as the angle approaches zero.

Radian Measure

Calculations in calculus, especially those involving trigonometric functions and their limits or derivatives, require angles to be measured in radians. Radian measure ensures that formulas such as the derivative of cosine or the Taylor series approximations hold true.

Numerical Methods for Evaluating Limits

Using calculators to compute function values at inputs close to a point of interest allows for practical approximations of limits. This numerical approach complements analytical methods by providing empirical evidence of the behavior of functions as variables approach specific values.

Transcript

00:01 In this video, we are going to use our calculator in radian mode to make some conclusions and find some values about a cosine function.

00:13 So the first thing i want you to do is make sure your calculator is in radian mode.

00:18 Some of you will have a button on your calculator that has something that looks like this, like degree and then arrow to radian.

00:28 It might be the other way around.

00:30 Press that button and make sure the word radian is highlighted or showing up.

00:38 If you have a t .i.

00:42 Calculator, graph and calculator, you're going to want to hit the mode button.

00:46 Go to the third row down and make sure you use your cursor to highlight the word radian and push.

00:56 Okay, so let's go through.

00:58 I'm going to go through the first two show you how to plug those in.

01:02 And then i'm going to pause the video and i want you to fill in the rest of the table yourself using the methods that i show you.

01:11 So if i want to find when theta is 0 .5, what will my output be? you go to your function and in your calculator you're just going to type the cosine button, plug in 0 .5 for theta, and then minus 1.

01:30 So if i hit my cosine button, 0 .5, close the parentheses, then minus 1.

01:38 I get a negative decimal.

01:39 I get negative 0 .124, and it keeps going.

01:52 So then to do the next one, instead of plugging in 0 .5, i'll plug in 0 .4.

01:57 Same equation, same buttons.

02:00 You're just going to do 0 .4 instead of 0 .5.

02:03 In this case, i get a negative .07, 8, 9, and it continues.

02:19 One thing that i do want to help you out with is skip down to this one right here.

02:24 The 0 .01.

02:27 Let's look at that.

02:28 So if you do cosine of 0 .01 minus 1.

02:32 If you type that in your calculator, you might get some weird notation.

02:37 So in my calculator, i am getting negative 4 .19 .9, so on.

02:46 And then there's like more numbers after that.

02:48 And then i have a little e with a negative 5.

02:53 So this is scientific notation.

02:55 This is kind of like times 10 to the negative fifth power.

02:59 If you wanted to write it in decimal form, that would mean you would have to scoot your negative over.

03:07 And you'd have to move the decimal 5 times.

03:09 Times to the left.

03:11 So 1, 2, 3, 4, 5, the decimal goes right there and we fill in these blanks with zeros.

03:19 So this would really be like a negative point, 1, 2, 3, 4 zeros, and then a 4, which since after it's a 9, i'll put a 5.

03:31 All right.

03:33 So if you see any notation like this, that is the same thing as scientific notation.

03:38 You can keep it in scientific notation or you can write it in decimal form.

03:45 Take a minute and fill in the rest of the table...

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