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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.$$f(x)=x^{3}+3 x^{2}-9 x-15$$

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 3

Concavity and the Second Derivative

Derivatives

Campbell University

Oregon State University

Harvey Mudd College

Baylor University

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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Use the First Derivative T…

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Question 50 is asking you to use the 1st and 2nd derivative to sketch the graph of F of X equals X to the third plus three X squared minus nine X minus 15. Um, you can also use intercepts if they're easy to find. So starting off with F prime of X, which will give us our critical points. That would be three X squared plus six X minus nine. Ah, you can go ahead and factor this so three times X squared plus two x minus three would just be three times x plus three times X minus one. So are critical points are at X equals negative three and one. Plugging those into a VAX. You would get negative 3 20 from negative 3 12 and one negative 20 from there, taking F double prime of X so we can find out if these are Max or men's would be six X plus six. So plugging in those points F double prime of three is equal to negative 12, which is concave down. A maximum F double prime of one is equal to 12. So concave up or a minimum and one ah, intercept, you can use is f of zero is equal to negative. 15 Negative. 15 zero. Um, And over here, Negative three and 12. So putting in those points. So negative 3. 12, But here than one negative 20 Somewhere down here, um, going ahead and connecting those points. You know that at negative 3. 12, you have a maximum. So it should look something like this you're crossing over here, and this is a minimum where it goes back up, and that is your answer to question.

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