Find the critical numbers of the function. h(t)=t^3 / 4-2 t^1 / 4

Name: Problem 37, Chapter 4: Calculus Early Transcendentals
Uploaded: 2020-05-06T15:44:58-08:00
Duration: 4 min 53 s
Channel: Chen Chen
Description: Find the critical numbers of the function. h(t)=t^3 / 4-2 t^1 / 4

Find the critical numbers of the function. h(t)=t^3 / 4-2...

Key Concepts

Critical Numbers

Critical numbers are points in the domain of a function where its derivative is zero or undefined. They are important in determining where the function may have a local maximum or minimum. These numbers help in the analysis of the function's behavior by identifying potential turning points or points of inflection.

Differentiation

Differentiation is a process in calculus used to determine the rate at which a function is changing at any given point. It provides the slope of the tangent line to the function at any particular point and is fundamental in finding critical numbers and analyzing the function's behavior.

Power Rule for Differentiation

The power rule is a basic technique for differentiation which states that if a function is expressed as f(x) = x^n, then its derivative is f'(x) = n*x^(n-1). This rule is particularly useful for functions with power expressions, including those with fractional exponents.

Fractional Exponents

Fractional exponents represent roots and powers in a compact form, where an exponent like a/b signifies the b-th root of a number raised to the power a. Understanding fractional exponents is crucial for differentiating functions that contain radical expressions, as they allow the application of the power rule.

Setting the Derivative Equal to Zero

Setting the derivative equal to zero is the process of finding potential critical numbers by solving the equation f'(x) = 0. This step is used to locate points at which the function's slope is zero, indicating potential local extrema or points of inflection, which is essential in understanding the overall behavior of the function.

Transcript

00:01 All right, everyone, we're gonna be taking a look at the critical values of the function given here.

00:06 And in order to do that first, let's understand what a critical value is.

00:09 So if we visualize what the function actually looks like on a why and on x axis function looks something like that, um and so critical value is their points where a mathematician would want to analyze when study angle function.

00:25 And those generally are the points at which the first derivative or the rate of change of the function, becomes zero or undefined.

00:32 One example is something that you see here.

00:34 It's a local minimum.

00:36 It's like a turning point for the function where the function goes from decreasing to increasing.

00:40 So that's an example of a critical value.

00:43 So, in order to find the values of t in which the first derivative is zero are undefined first only to find the first derivative.

00:53 So the first year of it ever gonna find using the power rule, which means that the exponents you bring the exponents down to become the coefficient.

01:03 Since the first time, the coefficient is originally one.

01:06 Anything multiplied by one stay the same so that the coefficient becomes three perform, which was the exponents.

01:12 And then you subtract one from the co from the exponents.

01:16 Rather, so we're going to do that for the second term as well.

01:20 Um, since there isn't existing coefficients there, we needed to know multiply backed by the value of the exponents in the original function and then subtract one from that exponents, which becomes, um, negative 3/4.

01:34 So in order to find the critical values again, their points at which the first derivative is equal to zero worth fine.

01:40 So let's do that by setting the, um, first derivative to zero.

01:52 So that would be zero.

01:55 Um, so right away, we know that if t is equal to zero, then the first irrevocable equal zero because both terms contained zero.

02:03 Anything said multiplied by zero...

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