(a) If G(x)=4 x^2-x^3, find G^'(a) and use it to find equations of the tangent lines to the curv

(a) If G(x)=4 x^2-x^3, find G^'(a) and use it to find...

(a) If $G(x)=4 x^{2}-x^{3},$ find $G^{\prime}(a)$ and use it to find equations of the tangent lines to the curve $y=4 x^{2}-x^{3}$ at the points (2,8) and (3,9) .
(b) Illustrate part (a) by graphing the curve and the tangent lines on the same screen.

Key Concepts

Differentiation

Differentiation is the process of finding the derivative of a function, which represents the function's instantaneous rate of change. In the context of polynomial functions, it involves applying rules such as the power rule to obtain a formula that can be used to calculate the slope of the function at any given point.

Tangent Line

A tangent line to a curve at a given point is a straight line that touches the curve without crossing it at that point. The slope of the tangent line is equal to the derivative of the function at that point, offering a linear approximation of the function near that point.

Point-Slope Form

The point-slope form is a method for writing the equation of a line when a point on the line and the slope are known. It is typically expressed as y - y? = m(x - x?), where m represents the slope and (x?, y?) is a point on the line.

Graphical Representation

Graphical representation involves visualizing both the original function and its tangent lines on the same coordinate system. This allows one to see the behavior of the function as well as how the tangent lines approximate the function at specific points, reinforcing the relationship between the derivative and the slope of the tangent line.

Transcript

00:01 Today we will be finding the equation of a tangent line to the curve y equals 4x square of minus x cubed at the points 2 8 and 3 9.

00:08 We will have two equations for this and to find the slope of these we'll use the equation m equals the limit as x approaches a of f of x minus f of a over x minus a.

00:19 We'll start with the point 2 8 and for this point 2 is our a.

00:24 So we'll start by pointing to the equation m equals the limit as x approaches 2 of 4.

00:30 4 x squared minus x cubed minus 4 times 2 squared minus 2 cubed divided by x minus 2.

00:44 That will equal the limit as x approaches 2 of 4x squared minus x cubed minus 2 minus 2 squared is 4 and 4 times 4 is 16 and 2 cubed is 8 over x minus 2 and this will equal limit as x approaches 2.

01:03 Of 4x squared minus x cubed minus 8 over x minus 2.

01:12 Factor 4x squared minus x cubed minus 8.

01:15 It will be the limit as x approaches 2 of negative x minus 2 times x squared minus 2 x minus 4 divided by x minus 2.

01:28 And the x minus 2 is cancel out.

01:31 So you're left with the limit as x approaches 2 of negative.

01:36 X squared minus 2x minus 4 and we can now plug in 2 for x so that will give us negative 2 squared minus 2 times 2 minus 4 and to simplify that gives us negative 4 minus 4 minus 4 and that 4 minus 4 cancels out so we are left with 4 that's the slope to find the equation of a line we'll use the equation y minus y0 equals m times x minus x 0 and we were given x0 y 0 in the beginning as 2 8 and we just found m to be 4 so to plug in we have y minus 8 equals m times sorry not m 4 times x minus 2 and to solve for y we'll get just y equals 4 equals 4x 4x as our first equation.

02:40 Now for the second equation, we will again use this.

02:42 We have m equals the limit as x approaches.

02:47 Go back to the question...

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