Consider y=x^3-1. (a) Sketch its graph as carefully as you...

Consider $y=x^{3}-1$. (a) Sketch its graph as carefully as you can. (b) Draw the tangent line at (2,7). (c) Estimate the slope of this tangent line. (d) Calculate the slope of the secant line through (2,7) and $\left(2.01,(2.01)^{3}-1.0\right)$. (e) Find by the limit process (see Example 1 ) the slope of the tangent line at (2,7).

Consider $y=x^{3}-1$.
(a) Sketch its graph as carefully as you can.
(b) Draw the tangent line at (2,7).
(c) Estimate the slope of this tangent line.
(d) Calculate the slope of the secant line through (2,7) and $\left(2.01,(2.01)^{3}-1.0\right)$.
(e) Find by the limit process (see Example 1 ) the slope of the tangent line at (2,7).

Key Concepts

Graphing Polynomial Functions

Graphing polynomial functions involves plotting the curve represented by a polynomial equation on the coordinate plane. It requires understanding the end behavior determined by the leading term, identifying intercepts where the graph crosses the axes, and finding turning points. For a cubic function, the graph typically has an 'S' shape with inflection points, indicating a change in concavity.

Tangent Lines

A tangent line to a curve at a given point is a straight line that touches the curve at that point without crossing it at that vicinity. It represents the instantaneous rate of change of the function at that point and has the same slope as the curve does at the point of contact.

Secant Lines

A secant line is a line that intersects a curve at two distinct points, and its slope represents the average rate of change of the function between those two points. It serves as an approximation to the tangent line as the interval between the points decreases.

Limit Process

The limit process is a fundamental concept in calculus used to define the derivative. It involves taking the limit of the difference quotient, which is the ratio of the change in the function's value to the change in the independent variable as the latter approaches zero. This process gives the exact slope of the tangent line at a point.

Derivative

The derivative of a function is a measure of how the function changes with respect to its variable. It is defined as the limit of the difference quotient and corresponds to the slope of the tangent line to the graph at any given point, providing valuable information about the function's instantaneous rate of change and behavior.

Transcript

00:01 I've started by drawing a graph in desmos, a piece of technology, and identified the point of interest.

00:08 That's the tangent point to seven.

00:13 So what can we do next? how about we try to draw the tangent line? so to do so, we're just going to sort of naturally extend the graph to be a line.

00:22 So i noticed that the graph itself is increasing quite rapidly at that point.

00:27 So if we tried our best to draw a tangent line, it would love.

00:31 Something like that, a pretty heavily steep graph.

00:38 So what is the slope of this line? well, we could sort of use the point 27 that we already have.

00:44 Another point on this line approximately is 1 negative 5.

00:49 So between these two points, let's calculate the slope.

00:54 The slope is the change in y, 7 minus negative 5 over the change in x 2 minus 1.

01:02 That's 12 over 1 or 12.

01:04 So we're estimating the tangent line here has a slope of approximately 12.

01:11 The next question asks, what can we do with a point very close to 27? so let's look at 27 and now 2 .01 and then 2 .01 plugged into the function.

01:27 That's its y value.

01:29 So now instead of estimating the tangent line, we're using the point 27 and a point really close to 2 .7.

01:35 That would be almost the blue point.

01:38 So we'd have 2 .01 cubed minus 1, minus 7, that's the change in y, and then we need the change in x.

01:49 That's 2 .01 minus 2 .2.

01:53 2 .01 cubed is almost 8.

01:58 In fact, it is 8 .1210601.

02:05 And if we subtract 1 and subtract 7, there we're essentially subtracting eight, and the change in x is 0 .01...