For each function, find (a) the equation of the secant line through the points where x has the g

For each function, find (a) the equation of the secant line...

Key Concepts

Secant Line

A secant line for a function is a straight line that passes through two distinct points on the graph of the function. It represents the average rate of change of the function between those two points. In mathematical terms, the slope of the secant line is given by the difference quotient, which involves calculating the difference in the function's output divided by the difference in the inputs.

Tangent Line

A tangent line to a function at a specific point is a straight line that just touches the curve at that point and has the same slope as the function does at that point. This line provides a linear approximation of the function near that point and is found using the derivative of the function evaluated at the point of tangency.

Derivative (Instantaneous Rate of Change)

The derivative of a function at a given point represents the function's instantaneous rate of change at that point, which is the slope of the tangent line. It is obtained through the limit process, where the difference quotient approaches an infinitesimally small interval, providing the exact slope at a specific point on the curve.

Average Rate of Change

The average rate of change of a function between two points is calculated by taking the difference in function values at those points and dividing by the difference in the corresponding x-values. This concept is directly linked to the secant line, as the slope of the secant provides this average change over the interval.

Equation of a Line

The equation of a line can be determined using the point-slope form, which requires a point on the line and the slope. For both secant and tangent lines, identifying a point through the function evaluation and computing the appropriate slope are essential steps before writing the final linear equation.

Transcript

00:01 So the function is fx is x squared plus x with two given x values x is 3 and x is 6.

00:07 We have to find equation of a second line at the given x values and equation of a tangent when x is 3.

00:15 So before we start this question, let's quickly go where if i have to find equation of any line, what do i need? if i have a equation which is of this form, y minus y1 equal m x minus x1.

00:32 I need a point x1 y1 and i need a slope.

00:38 So at least we need this.

00:40 And this before again we start the problem.

00:43 What's the meaning of second line? seekind line is, for example, if you have a graph, and a graph.

00:56 And there's a line passing through this.

00:59 The line that intersect this graph at two points is called a second line.

01:05 So in order to find equation of this line, i need slope.

01:08 For that, i need these two points.

01:11 So let's first figure out the points.

01:13 So when x is three, so we are given two points, three and six.

01:17 So when f x is three, it will become three squared plus three, which is nine plus 312.

01:29 When x is 6, i plug in 6 here, so 6 squared plus 6, 36 plus 6 is 42.

01:41 So we have 2 points, 312 and 642.

01:52 So basically we have two points.

01:55 And to find the slope, we have to find the rise overrun.

02:01 That means change in x value divided by change in change in y value divided by change in x value so slope is given by let me write the formula first of a minus fb minus f a where a f a is a point b f b is a point so f b let's pick so let's pick the first b value to be 42 that is why minus 12 is another y value divided by 6 over 3...

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