Find the average rate of change of the function over the given interval or intervals. h(t)=cott

Find the average rate of change of the function over the...

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Trigonometric Function (Cotangent)

Cotangent is a trigonometric function defined as the ratio of the cosine to the sine of an angle. Understanding its properties, such as periodicity, discontinuities, and its behavior over specific intervals, is crucial when analyzing changes in functions that involve trigonometric ratios.

Average Rate of Change

The average rate of change of a function over an interval measures how much the function value changes per unit change in the input. It is computed using the difference quotient, (f(b) – f(a))/(b – a), and represents the slope of the secant line connecting the points (a, f(a)) and (b, f(b)).

Difference Quotient

The difference quotient is the formula used to determine the average rate of change between two points on a function. It forms a foundational concept in calculus as it leads to the definition of the derivative, which measures an instantaneous rate of change.

Transcript

00:01 So on this question, we're going to look at finding the average rate of change when the interval is or when the function is cotangent of t.

00:09 And for the first interval, let's take a look at going from pi over four to three pi over four.

00:16 So the average rate of change going from pi over four to three pi over four.

00:24 For average rate of change, we want to use f of b minus f of a over b minus a.

00:31 In this case, we're working with a function h.

00:34 So you can just change that to h of b and h of a.

00:39 And then b and a are the end points on the interval.

00:43 So in this case, h of b is h of 3 pi over 4, minus h of pi over 4.

00:51 And then i'm going from b to a.

00:53 So minus on the denominator, 3 pi over 4, minus pi over 4.

00:59 So now let's take a look at some of those values.

01:02 So h of 3 pi over 4 is cotangent at 3 pi over 4.

01:06 Remember that's where sign and cosine are the same, but they have opposite sign.

01:11 So that's a negative 1.

01:13 And then h of pi over 4 is cotangent of pi over 4, and that's where they are equal to 1, where sign and cosine are equal.

01:20 And then on the denominator, you're going to have 3 pi minus 1 pi, which makes it 2 pi over 4...

Please give Ace some feedback