In Exercises 93–98, let
f(x) = sin x, g(x) = cos x, and h(x) = 2x...

Question

In Exercises 93–98, let
f(x) = sin x, g(x) = cos x, and h(x) = 2x.

Find the exact value of each expression. Do not use a calculator.
the average rate of change of f from x₁ = 5𝜋/4 to x₂ = 3𝜋/2
(Hint: the average rate of change of f from x₁ to x₂ is f(x₂) - f(x₁).)
x₂ - x₁

Accepted Answer

Welcome back. I am so glad you're here. We're asked to determine the exact value of the average rate of change for the given function from X sub one equals nine pi divided by four, two, X sub two equals five pi divided by two without the help of a calculator. And then we're given that F of X equals the cosine of X. Our answer choices are answer choice A, the squared of two divided by pi answer choice, B negative two square root two divided by pi answer trace C two square at two divided by pi and answer choice D negative square root two divided by pi. All right. So we recall from previous lessons that the average rate of change for functions is given to us by this fraction. In the numerator, we have F of X sub two minus F of X sub one. And that's divided by, in the denominator X sub two minus X sub one. And we know what some of these are, we know what are X sub two and our X sub one are. So we could fill those in X sub two that five pi divided by two. Then minus our X sub one which is nine pi divided by four. But what about our F of X sub two and our F of X sub one? Well, that's where we use our function. We've got F of X is the cosine of X. So for both X sub one and X sub two, we'll be plugging those in to our function. So F of X sub one is going to be the cosine of X sub one, which is nine pi divided by four. And F of X sub two is going to be the cosine of X sub two which is five pi divided by two. And we can evaluate those. Let's start with the cosine of nine pi divided by four. So we know that if we have nine pi divided by four, that's more than one full rotation around a circle. So we can take our nine pi divided by four and subtract rotation, subtract a two pi but we've got a denominator of four. So for subtracting two pi with the denominator of four, that's going to be eight pi divided by four and nine pi divided by four minus eight pi divided by four is just pi divided by four. And there we go, that's on our unit circle. If we draw a rough sketch of a coordinate plane, we've got a horizontal, well, that's not even very horizontal. We've got a horizontal X axis, a vertical Y axis, they come together at the origin in the middle. Now drawing our angle, Pi divided by four. The initial side has a vertex at the origin and heads toward positive infinity along the X axis. And the terminal side pi divided by four is exactly halfway between the positive part of the X axis and the positive part of the y axis in the first quadrant, that's our pi divided by four in quadrant one. And so pi divided by four in the first quadrant, we don't need a reference angle. In the first quadrant, we recall from previous lessons that pi divided by four is 45 degrees. And so knowing that it's 45 degrees, we can draw a rough sketch of a 45 45 triangle, 90 degrees is the right angle. The other two are both 45 degrees. And recalling the side lengths for this, the hypotenuse is the square root of two and both the opposite and adjacent are one. So if we're talking about the cosine of degrees, back to that cosine of 45 degrees, we know that the cosine is going to be the hypo or excuse me, not the hypotenuse or cosine, it's adjacent, divided by hypo. And here the adjacent is one divided by the hypotenuse, which is squared of two. And then very quickly, we don't want the squared of two in the denominator. So multiplying this fraction by squared of two divided by the squared of two we have the squared of two in the numerator divided by two in the denominator. So our F of X sub one, the cosine of nine pi divided by four is the square root of two, divided by two. And now we want to determine the cosine of five pi divided by two. So for this one, we know that five pi divided by two is going to be more than one full rotation. So we can subtract a two pi but this time we've got a denominator of two. So that's going to be four pi divided by pi minus four. Pi is just one pie. We have pi divided by two but pi divided by two, that's the positive part of our Y axis. This is a quadrant angle. Pi divided by two is our 90 degree angle. And we recall from previous lessons that the cosine of pi divided by two is equal to zero. So we've got F of X sub two is zero. So in our numerator, we've got zero minus square at two, divided by two. That's divided by, in the denominator, five pi divided by two minus nine pi divided by four. Now we just need to do some simplification here. So in our denominator, we need a common denominator within the denominator to subtract. So we can multiply the five pi divided by two by two, divided by two. So we'll have in that denominator 10 pie divided by two minus nine or excuse me, divided by 4, 10 pi divided by four minus nine pi divided by four, which will give us just one pi divided by four. So now I'm going to write this over here. We have our zero minus squared of two divided by two. That's just the negative square at two divided by two. And then I'm going to write that with a division sign for our second fraction divided by pi divided by four because we know that when we are dividing two fractions, we're going to multiply by the reciprocal of the second. So we have negative square at two divided by two multiplied by the reciprocal of pi divided by four, which is four divided by pi. And now we can simplify this, the four in the numerator reduces the two in the denominator to just a two in the numerator. So we've got in our numerator, we've got two multiplied by the squared of two and that's negative divided by pi and that is our answer for this one. We look at our answer choices and that matches with answer choice. B well done. We'll catch you on the next one.

In Exercises 93–98, let f(x) = sin x, g(x) = cos x, and h(x) = 2x. Find the exact value of each expression. Do not use a calculator. the average rate of change of f from x₁ = 5𝜋/4 to x₂ = 3𝜋/2 (Hint: the average rate of change of f from x₁ to x₂ is f(x₂) - f(x₁).) x₂ - x₁

Key Concepts

Average Rate of Change

Trigonometric Functions

Radians and Angle Measurement

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