For the functions in Problems: (a) Find the derivative at...

For the functions in Problems: (a) Find the derivative at $x=-1$ (b) Find the second derivative at $x=-1$ (c) Use your answers to parts (a) and (b) to match the function to one of the graphs in Figure $3.9,$ each of which is shown centered on the point (-1,-1) . $$f(x)=2 x^{3}+3 x^{2}-2$$

For the functions in Problems:
(a) Find the derivative at $x=-1$
(b) Find the second derivative at $x=-1$
(c) Use your answers to parts (a) and (b) to match the function to one of the graphs in Figure $3.9,$ each of which is shown centered on the point (-1,-1) .
$$f(x)=2 x^{3}+3 x^{2}-2$$

Key Concepts

Derivative

Derivative represents the instantaneous rate of change of a function with respect to its independent variable. It gives the slope of the tangent line to the function at any particular point, and it is found by applying rules such as the power rule, product rule, quotient rule, or chain rule. This concept is fundamental for understanding how a function behaves locally, and it facilitates analyses such as identifying increasing or decreasing intervals.

Second Derivative

The second derivative is the derivative of the derivative of a function. It provides information about the curvature or concavity of the function, indicating whether the function curves upwards (concave up) or downwards (concave down) at a given point. It is also closely related to the concept of acceleration in physical applications and can be used to identify points of inflection where the concavity changes.

Graphical Analysis Using Derivatives

Graphical analysis using the first and second derivatives involves matching the slope and curvature properties at a point with the visual appearance of a graph. The first derivative indicates the slope of the tangent at a point, while the second derivative indicates the concavity. By comparing these values, one can determine which graph among several options best represents the function’s local behavior around a specific point, such as identifying the correct orientation and curvature of the graph centered at that point.

Transcript

00:01 We're given this function f of x, and for part a of our question, we're going to try to find the derivative of our function at x is equal to negative 1.

00:10 And so the first thing that we're going to do is find the derivative of our function.

00:15 So f prime of x is equal to...

00:19 We're going to use the power rule to find this derivative.

00:23 And what we do is we take down the exponent, and we multiply it by our x and our whatever x is multiplied.

00:30 By it in this case two, and then we minus 1 to our exponent.

00:34 So we're going to have 3 times 2, which is 6 times x to the 3 minus 1 exponent, which is 2.

00:42 And we're going to do the same thing for 3x squared, and in that case we get 6x.

00:47 And then as for our constant, when we take the derivative of a function, all constants go to 0.

00:53 So this negative 2 just becomes 0.

00:56 And now that we found this derivative of our function, we can find the derivative at negative 1, which is equal to 6 times negative 1 squared plus 6 times negative 1, which is equal to 6 minus 6, which is equal to 0.

01:13 So our derivative at f is equal to negative 1 is equal to 0.

01:20 And now for part b, we're going to find the second derivative of our function f, and how we're going to do that is by finding the derivative of our first derivative.

01:29 So f double prime of x is equal to the derivative of 6x squared plus 6x.

01:37 And we're going to use the power rule again to find this derivative...

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