Find equations of both lines that are tangent to the curve y=1+x^3 and are parallel to the line

Find equations of both lines that are tangent to the curve...

Key Concepts

Derivative

The derivative of a function gives its instantaneous rate of change at any given point, which is used to determine the slope of the tangent line to the curve at that point.

Tangent Line

A tangent line is a straight line that touches a curve at just one point, sharing the same slope as the curve at that point. It provides the best linear approximation of the curve near the point of tangency.

Parallel Lines

Parallel lines are lines in a plane that never intersect, which means they have identical slopes. When finding tangent lines that are parallel to a given line, one sets the derivative of the function equal to the slope of the given line.

Point-Slope Form

The point-slope form is an equation of a line expressed as y - y? = m(x - x?), where m is the slope and (x?, y?) is a point on the line. This form is useful for writing the equation of a tangent line once the point of tangency and the slope are known.

Transcript

00:01 Okay, let's say that we're trying to find two lines tangent to the curve y equals 1 plus x cute and parallel to the line 12x minus y equals 1.

00:25 Okay, so we're looking for two lines here, y1 and y2, we can call them, and we're going to write these in slope intercept form, which means we're looking for some slope for y, 1 m1 plus some y intercept for the first line we'll call it b1 and we'll do the same thing for y 2 m2 x plus b2 okay so we are given that these two lines are tangent to this curve and parallel to this line well what does this fact give us we know that parallel lines have the same slope so we need to figure out this slope of this line.

01:12 Let's go ahead and put this line in y intercept form.

01:16 I'm going to do that by moving this y over by adding it to both sides.

01:20 I get 12x equals 1 plus y, and then i'm going to move the 1 over by subtracting it.

01:27 So i have 12x minus 1 is equal to y.

01:32 So what i have here is that the slope of this line is 12.

01:37 And because we know that y 1 and y2 are parallel to this line that automatically tells me that y1 has a slope of 12 and y2 also has a slope of 12.

01:54 So we've already found both of our slopes.

01:57 Now we just need to find the y intercepts.

01:59 And to do that we're going to use this other piece of information, which is that these lines are tangent to the curve y plus y equals 1 plus x cubed.

02:10 Alright, so what does this tangent mean? well, we know that a slope of a tangent line is equal to the derivative.

02:29 So the slope of a tangent at a point is equal to the derivative at that point.

02:34 So we need to go ahead and figure out what our derivative is.

02:38 So to do that we're going to use two of our differentiation rules, the constant rule, which tells us that the derivative of a constant is zero.

02:46 So when we take the derivative of that one, we get zero.

02:51 Then we're going to use the power rule, which tells us to take the derivative of x cubed, we move the power down, so we get 3 as a coefficient, and then we drop the power by 1.

03:02 And so we get 3 minus 2, sorry, 3 minus 1 is 2, which gives me 3x squared.

03:09 So our derivative here is 3x squared.

03:15 Okay? we are looking for the point where these two lines are tangent to this curve, which means we want the slope, which is 12, to be equal to the derivative.

03:27 And so i'm going to go ahead and set those equal over here.

03:30 I've got 12 equals 3x squared.

03:34 And now i want to solve for x.

03:36 So i'm going to divide by 3.

03:38 That gives me 4 equals x squared.

03:42 And then i'm going to take the square root, which tells me that x is equal to plus or minus 2.

03:50 Remember when we take the square root, we get a positive answer and a negative answer...