(a) Use Definition (3.1) to find the slope of the tangent line to the graph of f at P(a, f(a)).

(a) Use Definition (3.1) to find the slope of the tangent...

Key Concepts

Limit

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain point. Understanding limits is crucial for defining the derivative, as it forms the basis of evaluating the instantaneous rate of change of a function by considering the behavior of the difference quotient as the interval shrinks to zero.

Difference Quotient

The difference quotient, given by (f(a+h) - f(a))/h, represents the average rate of change of a function over an interval of length h. It is a central concept in determining a derivative because taking the limit as h approaches zero transforms the average rate of change into the instantaneous rate of change at a specific point.

Derivative

The derivative of a function at a point is defined as the limit of the difference quotient as the interval approaches zero. It quantifies the instantaneous rate of change of the function, or equivalently, it provides the slope of the tangent line to the function's graph at that point.

Tangent Line

A tangent line to a function's graph at a particular point is a straight line that touches the graph at that point and has the same instantaneous slope as the function at that point. Its equation is often determined using the point-slope form once the derivative (i.e., the slope at the point) has been computed.

Point-Slope Form

The point-slope form is an equation of a line expressed as y - y? = m(x - x?), where m is the slope of the line and (x?, y?) is a point on the line. This form is particularly useful in finding the equation of a tangent line once the slope (derived from the derivative) and a point of tangency are known.

Transcript

00:01 For this problem, we have to find the slope of the tangent line to f at point p.

00:06 Using the definition in the red box here, which is m sub a equals the limit as h approach to 0, of f of a plus h minus f of a all over h.

00:16 So the first thing i want to do is start building my equation for m sub a by substituting a plus h into the f of x equation for every where i see x, and then also plugging in for a.

00:32 So m sub a equals to the limit as h approaches 0 of a plus h cubed minus a cubed all over h.

00:52 So now i have to multiply the a plus h cubed and that's going to give me the limit as h approaches zero of a cube plus 3 a squared h plus 3 a squared plus h squared plus h cubed minus a cubed all over h so if you look at the numerator you'll see that a cube does cancel out with this a cube here leaving us with the limit as h approaches 0 of 3a squared h plus 3ah squared plus hq h h h q, h.

02:05 Now plug it in for zero, that's going to, i'm sorry, not plugging in for zero, but first we have, we see that each term has an h that the h in the denominator can cancel out with.

02:23 So this h here, one of the h is here, and one of the h is here, counsels out with this h here.

02:33 So now the equation becomes the limit.

02:37 This h approaches 0 of 3a squared plus 3ah plus h squared.

02:53 Now i can plug in for 0, and that gives me 3a squared because both of these cancel out due to the h being 0.

03:07 And that is my slope.

03:20 So the next part of this problem asks us to find an equation of the tangent line at the point p, which is at 2f2 for the equation f of x equals x cubed.

03:35 So the first thing i'm going to do is i'm going to find my y value for the point p by substituting in the 2 for x.

03:46 So f of 2 equals to 2 cubed, which equals to 8.

03:54 So my y value is 8...

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