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Use the first and second derivatives to sketch the graph of the given equation. Also include the intercepts, whenever they are easily determined.$$f(x)=x \sqrt{x+4}$$

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 3

Concavity and the Second Derivative

Derivatives

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Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

30:01

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (the rate of change of the value of the function). If the derivative of a function at a chosen input value equals a constant value, the function is said to be a constant function. In this case the derivative itself is the constant of the function, and is called the constant of integration.

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Use the first and second d…

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we want to graph the function of X equals X times. Spirited experts for to do several deployed calculus curve setting techniques analyzing F. F. Prime and F double prime successively. So for F F we get basic info on the graph so the domain is negative for stability. The Y intercept X equals negative 40 and there are no horizontal vertical after jokes. F prime. Tell us where F is increasing or decreasing. F prime is three X plus eight over to root, expose four from the numerator drama equals zero. We get partitioning critical points at negative for negative eight thirds, so between negative four negative eight thirds, F prime is negative. So if it's decreasing to the right of negative eight thirds prime is positive. So it is increasing. Finally we analyze F double prime for cavity purposes. S prime is three X plus 16/4 times X plus 4 to 3 halfs. You can get critical points at negative four. And this time for the new year negative 16/3. We find that for all, acceptable prime is positive. So we have our ability to graph. So we have that F is defined from negative infinity, decreasing until negative eight thirds and then increasing through its intercept at X equals zero equals zero, up until infinity.

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