Find an equation for the tangent to the curve at the given point. Then sketch the curve and tang

Find an equation for the tangent to the curve at the given...

Key Concepts

Tangent Line

A tangent line to a curve at a given point is a straight line that touches the curve only at that point (locally) and has the same instantaneous slope as the curve. It provides a linear approximation of the function at that point and is a fundamental concept in calculus and differential geometry.

Derivative

The derivative of a function represents the rate at which the function value changes with respect to changes in its input, and it gives the slope of the curve at any specific point. In the context of finding a tangent line, the derivative evaluated at the point of interest provides the slope of the tangent.

Point-Slope Form

The point-slope form is an equation of a line that is especially useful when you know a point on the line and its slope. It is given by y - y? = m(x - x?), where (x?, y?) is the point on the line and m is the slope. This form is commonly used to write the equation of the tangent line using the derivative for the slope.

Curve Sketching

Curve sketching involves drawing a graph of the function to visualize key characteristics such as intercepts, curvature, and asymptotic behavior. When combined with the tangent line, it provides insights into the local behavior of the function at a specific point, illustrating the concept of linear approximation within the broader context of graphical analysis.

Transcript

00:12 Okay, so the problem indicates that we have to find the tangent line to the graph y equals x cubed at the point, at the point negative 2, negative 8.

00:27 Now, the first question that we have to ask ourselves is, what is a tangent line? and if you'll recall, the tangent line is a line that intersects this function at a single point, this point right here.

00:41 So the first thing that we need for a line is we need to figure out what the slope of that line is.

00:48 So naturally, that's the derivative.

00:50 So we're going to differentiate this function with respect to x, and we're going to get something that i call the slope function.

00:59 The slope function, the derivative of this, gives us the slope at any point, at any x value on this curve.

01:06 Specifically, we want the slope of this line at x equals negative 2.

01:12 So we're going to plug that guy in, so at x equals negative 2.

01:20 Okay, this is just symbolic notation for what i'm saying.

01:23 I want to find the slope at x equals negative 2, and that's going to equal three times negative 2 squared.

01:32 And actually calculating that you get 12.

01:38 Now the other thing that you need for a line is either a point or the y intercept.

01:44 Here it's easier to use point slope form, which you'll recall from algebra, is the formula y minus y1 equals m which is the slope x minus x1 the reason that's easy is because it gives you the point directly if it only gave you the x value you would have to plug that x value in the function to get the y value corresponding to this ordered pair okay so this is going to be your x1 and this is our y1 the subscript 1 just denotes that it's a specific point on this curve.

02:28 In fact, it's the only guaranteed point on this curve that also belongs to this function.

02:34 It's called the point of tangency.

02:36 So now that we have the slope, which we just computed by taking the derivative evaluated at the x value, and we have a point, we can easily plug it in for the equation of this line.

02:46 So we're going to get y minus y1 is going to be negative 8.

02:53 Make sure that you put this in parentheses because a common mistake is that people will ignore this and then they'll just write y minus 8 this negative sign is already embedded into point slope form so make sure you don't forget that okay and that's going to equal the slope 12 times x minus x1 so again that's a negative value make sure that you write two negative signs so you know to cancel them out for a positive okay so this is going to turn into y plus plus 8 equals 12 times x plus 2.

03:36 Okay, and sometimes this can be a little inconvenient, although this is a correct equation...

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