In Exercises 43-50 , consider the function on the interval (0,2 π) . For each function, (a) find

In Exercises 43-50 , consider the function on the interval...

Key Concepts

First Derivative Test

The First Derivative Test is a method used in calculus to identify local extrema of a function by analyzing the sign changes of its derivative around critical points. When the derivative changes from positive to negative at a point, it indicates a relative maximum, and a change from negative to positive indicates a relative minimum.

Critical Points

Critical points are values in the domain of a function where its first derivative is zero or undefined. They are important because they represent potential locations of local maxima, minima, or other significant changes in the behavior of the function.

Increasing and Decreasing Intervals

Increasing and decreasing intervals are determined by the sign of the first derivative. If the derivative is positive on an interval, the function is increasing there; if the derivative is negative, the function is decreasing. This concept helps in understanding the overall behavior of the function on its domain.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables where the functions are defined. They are used to simplify expressions and make the process of differentiation or integration more straightforward, particularly when dealing with products of sine and cosine functions.

Graphical Verification

Graphical verification involves using graphing utilities or plotting software to visually confirm analytical results. This approach is useful for checking the accuracy of critical points, intervals of increase or decrease, and the overall shape of the function.

Transcript

00:01 So we're looking at this function f of x here, and we want to apply the first derivative test to find any relative minimums or maximums along this open interval from 0 to 2 pi.

00:12 And the first derivative test we can apply to the values of f prime of x along certain intervals determined by the critical numbers.

00:22 And the critical numbers are found by taking the derivative of f of x and solving for 0, solving x when the function is equal to zero.

00:31 So if we find that we have f prime of x along those intervals going from negative to positive, we have a relative minimum.

00:39 And if we have f prime of x going from positive to negative, that's a relative maximum.

00:44 So what we first have to do is we have to get the derivative of f of x.

00:49 Now we can't directly take the derivative of sine x times cosine x.

00:54 But what we can do is we can use the fact that we know that sine x times cosine x is equivalent to sine of 2x divided by 2.

01:07 And so this is a function that we can take the derivative of.

01:11 And we do this by applying the chain rule.

01:14 So we'll have this function, this being f of x, rewritten.

01:21 And if we're to go ahead take the derivative, what we then end up getting is co -sounding.

01:33 2x.

01:38 So now that we have our derivative function, f prime of x, we're going to go ahead set that equal to 0.

01:48 And when we do that, we'll find a number of different values that satisfy that equation.

01:54 So going from the smallest to the largest, we have pi over 4, 3 pi over 4, 5 5 pi over 4 and 7 pi over 4.

02:20 So now we found our critical numbers and what we want to do is we want to set up the intervals that we're looking at the value of f prime of x 4.

02:29 So remember that we're looking at the open interval from 0 to 2 pi.

02:36 So that's the total amount of the interval that we're looking at.

02:42 So we have first 0 to pi of 4, and then each interval is just going to go between each of these critical numbers till the end of the open interval that we're looking at.

03:01 So we have from pi over 4 to 3 pi over 4, 3 pi over 4 to 5 pi over 4, 5 pi over 4 to 7 pi over 4.

03:22 5 over 4 to 7 pi over 4.

03:29 And then finally we have 7 pi over 4 to 2 pi.

03:43 So now what we want to do is we want to look at the sign of f prime of x.

03:48 So this means taking a value between within each of these test intervals and determining the sign...

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