The tangent line to the curve y=x^3-6 x^2-34 x-9 has slope 2 at two points on the curve. Find th

The tangent line to the curve y=x^3-6 x^2-34 x-9 has slope 2...

Key Concepts

Polynomial Differentiation

When dealing with polynomial functions, differentiation is straightforward due to the power rule, which simplifies finding the derivative. The resulting derivative function is then used to solve for the x-coordinates where the tangent line meets a certain slope.

Equating the Derivative to a Given Slope

This concept involves setting the derivative of a function equal to a specific slope value to determine the points at which the curve has that slope. It connects the algebraic process of differentiation with geometric interpretations of the curve.

Tangent Line

A tangent line to a curve is a straight line that touches the curve at one point and has the same slope as the curve at that point. It is used to approximate the behavior of the curve near the point of tangency.

Derivative

The derivative of a function represents the instantaneous rate of change at any point. It is computed using differentiation rules, such as the power rule for polynomials, and it gives the slope of the tangent line to the graph of the function at a specific point.

Transcript

00:01 If we're given the curve y equals x cubed minus 6x squared minus 34x minus 9, we want to find out at what points is the slope of the tangent line to this curve equal to 2.

00:22 The key idea that we want to remember is that the slope of the tangent line at the point x, or let's use different notation, let's use at the point.

00:51 A, f of a, to the curve y equals f of x, is the derivative at that point.

01:06 So is f prime of a if it exists.

01:20 To draw a picture of this idea, let's say this is our curve, y equals f of x.

01:39 And here's our point.

01:40 So x value is a, y value is f of a here's my tangent line at that point so the statement is that the slope is equal to the derivative evaluated at a so that means what we want to do for this problem is we want to first find the derivative of our curve.

02:31 So find y prime, which is f prime of x.

02:39 Then we want to find where f prime of x is equal to two.

02:48 Since we want the slope of the tangent line to be equal to two, since the slope of the tangent line is the value of the derivative, we need to set f prime of x equal to two.

03:07 And then the third step would be simply plug in the solutions to part two to your equation for your curve, and in that way we'll find the y coordinate.

03:36 So let's go through these steps.

03:38 I'm going to rewrite my curve, the equation for my curve.

03:51 We're going to take the derivative, y prime, is equal to 3x squared minus 12x minus 34.

04:02 We can also use the notation.

04:05 This is what we're calling f prime of x.

04:10 So that's step one.

04:11 We find our derivative.

04:15 Step two.

04:18 We're going to set f prime of x equal to 2.

04:25 3x squared minus 12x minus 34 is equal to 2.

04:30 And this is what we need to solve.

04:32 We want to solve for x.

04:33 So if we think back to, if we think of our algebra skills, we see that the highest degree of x is 2.

04:51 So we can transform this into a quadratic equation.

04:54 And to do that, we just need to subtract two from both sides, 3x squared minus 12x minus 36 equals 0.

05:05 So remember, a quadratic equation is just where you have a quadratic polynomial, aka a degree to polynomial set equal to zero.

05:16 And now we can solve.

05:18 So this is a quadratic equation where we can solve it by factoring.

05:26 So i'll first factor out of three from everything.

05:33 And then if i look at what's inside the parentheses, this is going to factor as x minus six times the quantity x plus two, since negative six times two is equal to negative 12, negative 6 plus 2 is equal to negative 4.

05:53 So the left -hand side is equal to 0 if x is equal to 6 or x is equal to negative 2.

06:05 So we know if these are the x values, if we know that for these two x coordinates, the derivative is equal to 2...

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