In Exercises 5–10, find an equation for the tangent line to th...

Question

In Exercises 5–10, find an equation for the tangent line to the curve at the given point. Then sketch the curve and tangent line together.

y = (1 / x²), (−1, 1)

Accepted Answer

Find the equation of the tangent line to the curve at the specified point. Also plot both the curve and the tangent line on the same coordinate plane. Where we're given, Y equals 4 divided by X2, and the point is 21. We're also given a coordinate plane to graph our equations on. Now, to solve this, we first need to find the derivative of this function. To do that, we'll make use of the difference quotient. This tells us that our derivative is equals to the limit. As H approaches 0, F X plus H. Minus F of X divided by h where F of X in our case. It's 4 divided by X squared. Now we'll simplify and plug in. We have F Of X plus H minus F of X, divided by H. So, we first have 4 divided by X plus H squared. Minus 4 divided by X2 all divided by H. That would simplify. We need to get the same denominator, so to do that, we will find the common denominator. We can multiply Our rightmost equation by X + H2 divided by X + H2. And our leftmost equation by X2 divided by X2. This gives us 4 X 2. Divided by X2, multiplied by X plus H2. -4, multiplied by X plus H2. Divided by X2 multiplied by X plus H squared. We can combine The two functions This is all divided by H. We don't want to forget that, but we can combine the two functions. To get 4 X squared. Minus 4 multiplied by X plus H2. All divided by X2 X plus H squared, which that in turn is divided by H. We can now simplify even further. We first need to know that X plus H2 expands to X2 + 2 XH plus H2, meaning we have 4 X2 minus 4 multiplied by X2 + 2 XH plus H2. Divided by X2, X plus H2, all divided by H. We can distribute I will draw an arrow to the left here. But we can distribute To get 4 X2 minus 4 X2. Minus 8 XH minus 4H squared. Divided by X2, X plus H2, all divided by H. We can cancel out our 4 X squads to get -8 XH minus 4H2. Divided by X2 X plus H2. All divided by H. Now, let's factor out an H. We have H multiplied. By negative 8X. Minus 4 H. Divided by X2 X plus H2, all divided by H. We can cancel out our Hs to end up with 8 X minus 4 H divided by X2 X plus H2. Now here we keep in mind that we had our limit from earlier. And we can now plug in our value of H being 0. You have negative 8 X minus 4 multiplied by 0. Divided by X2 X + 0 squad, which gives us -8 X. Divided by X to the 4th. That finally simplifies to -8, divided by X to the 3rd. We now have our derivative. Now, we are looking for at the point being 21. So we want F2. Which is -8 divided by 2 cuts. Or -8 divided by 8, which is -1. That means the slope of our tangent line is -1. Now, let's find our tangent lines equation. Do this, we can make use of point slope form. Where we have Y minus Y1 equals M multiplied by X minus X1. We have the point 21. And so we'll plug this into our equation. Why? -1 equals -1, multiplied by X minus 2. We can then simplify to get Y minus 1 equals negative X + 2. And if we add one Onto both sides, we get Y equals negative X plus 3. That is our tangent line. Finally, we can graph both of our functions. We are graphing The original curve and the tangent line. So our original curve. Has an asymptote at X equals 0. We can plug in some values defined points. And if we do that, we end up getting. Values such as 21. -14. And 1/16. We also get some more points reflected on the right side of the graph. 21 14 And 1/2, 16. We can finally draw a smooth curve. Between these points. We can make that a little better. Draw one more. Go up. And through our points. Like so We also need the graph of the tangent line. The tangent line Is Y equals x + 3, which is linear. One of the points is at 21, and the other one is at -14. We can plug this into our equation to confirm. But we end up getting. A straight line that goes to both of those points on a graph. OK, I hope to help you solve the problem. Thank you for watching. Goodbye.

In Exercises 5–10, find an equation for the tangent line to the curve at the given point. Then sketch the curve and tangent line together.

y = (1 / x²), (−1, 1)

Key Concepts

Derivative

Point-Slope Form

Sketching Curves and Tangent Lines

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