Use the definition of a derivative to find f'(x) and f”(x) . Then graph f , f' , and f” on a com

Use the definition of a derivative to find f'(x) and f”(x) ....

Key Concepts

Limit Definition of Derivative

This concept involves finding the derivative of a function by evaluating the limit of the difference quotient as the interval approaches zero. It is the foundational method for determining the instantaneous rate of change, ensuring that one fully understands the transition from average rates of change over an interval to the exact rate at a point.

First Derivative

The first derivative represents the slope of the function at any given point, indicating how the function is increasing or decreasing. It is crucial for identifying local extrema, understanding the linear approximation of functions, and analyzing the overall behavior such as intervals of monotonicity.

Second Derivative

The second derivative, being the derivative of the first derivative, informs us about the curvature or concavity of the function. It helps in determining whether the function's rate of change is increasing or decreasing, which is essential for identifying points of inflection and assessing the 'acceleration' of a change in the function.

Graphical Analysis of Functions and Their Derivatives

Graphing a function alongside its first and second derivatives allows one to visually confirm and interpret analytical findings. The relationship between these graphs aids in understanding how the function's slope evolves and how the concavity impacts the overall shape, ensuring that the calculated derivatives align with the visual behavior of the function.

Transcript

00:01 This is problem number 53 of the stewart calculus safe edition, section 2 .8.

00:05 Use the definition of a derivative to find f prime of x and f double prime of x.

00:09 Then graph f, f prime, and f double prime on a common screen, and check to see if your answers are reasonable.

00:15 So, let's start with f prime of x.

00:21 This is defined as the limit as h approaches zero of this function evaluated at x plus h.

00:33 3 times a quantity x plus h squared plus 2 times the quantity x plus h plus 1 minus the function by itself 3x squared plus 2x plus 1 okay and this is all divided by h our next step is to expand the numerator a little bit this biomile squared x plus h the x squared plus 2xh plus h squared multiplied by three will be 3x squared plus 6x h plus 3h squared then we're going to multiply and distribute the 2 here 2x plus 2h and then we have a plus 1 and then in this part here we're going to distribute the negative negative 3x squared negative 2x negative 1 and this is all divided by h.

01:47 So now we identify some terms that might be cancelled, such as 3x squared and negative 3x squared, positive 2x and negative 2x, and positive 1 and negative 1.

01:59 The remaining three terms, 6xh plus 3h squared plus 2h have 1h in common, so that can cancel with the agent denominator.

02:10 Therefore, we're left over with the limit as each approaches 0 of 6x4.

02:17 Plus 3h plus 2 and as h approaches 0 3h approaches 0 answer here is 3x plus 2 6x plus 2 and that is our answer for f prime of x let's do the same for the next derivative we're doing f double prime of x and we're using this function now the resultant derivative as the function to evaluate this limit with the limit of each approach to 0 we're doing the same steps again just with the new function.

02:53 The function is 6x squared plus 2 and we're going to plug in x plus h here and then we're subtracting 6x plus 2.

03:07 Put this in some parentheses, all divided by h.

03:14 Okay, the next step will be to distribute the 6 in the numerator 6x plus 6h plus 2 and then distribute the negative negative 6x minus 2 divided by h.

03:35 6x and negative 6 will cancel plus 2 and negative 2 will cancel.

03:39 We're left with 6h over h, which if we simplify becomes just 6, that's the only remaining term...

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