{Use of Tech} Equations of tangent lines
Find an equatio...

Question

{Use of Tech} Equations of tangent lines

Find an equation of the line tangent to the given curve at a.

y = −3x² + 2; a=1

Accepted Answer

Hi everyone, let's take a look at this practice problem dealing with tangent lines. This problem says to determine the equation of the tangent line to the curve at the specified point. And we're given for a curve, Y is equal to 4 X2 minus 5, and the point is at X equals to -2. We're also given 4 possible choices as our answers. For choice A, we have Y is equal to 16 X plus 21. For choice B, we have Y is equal to 16 X minus 21. For choice C, we have Y is equal to minus 16 X plus 21. And for choice D, we have Y is equal to minus 16 X minus 21. Now, in order to determine the equation for our tangent line, we'll first need to determine the slope of our tangent line at the point in question. So, for our slope, which we'll label as M, that's going to be equal to the derivative of Y with respect to X. Evaluated at a point, which is X equal to -2. So this is going to be equal to the derivative with respect to X. Of the quantity of 4 X2 minus 5. And that is evaluated at X equal to -2. So, taking the derivative, for the first term, we can use our constant multiple and power rule to take this derivative. So we'll multiply the coefficient by the exponent, so we'll have 4 multiplied by 2, and then that will be multiplied by X, raised to the 2 minus 1, so I need to subtract 1 from the exponent. Then I will have minus the derivative of 5 with respect to X, since 5 is a constant, and when we take a derivative of a constant, we get 0, so we'll have minus 0. And this whole quantity is going to be evaluated at X equal to -2. So we'll simplify this expression, so this is going to be equal to or multiplied by 2, which is 8, and that's gonna be multiplied by X and I'll have X to the power 1, which is just X. And then this is going to be evaluated at X equal to -2. So doing that evaluation, this is going to be equal to 8, multiplied by -2, which is equal to -16. So that is the slope of our tangent line. And now we need to write an equation for our tangent line. And one way we could write an equation for that line is to use our point slope form. So in order to use the point slow form, I'll need to find the Y value when X is equal to -2. So, we use our equation for Y that we're given the problem to calculate that. So we'll have Y equal to 4 multiplied by minus 2 squared. Minus 5. And when we evaluate this expression, this is going to be equal to 11. So now I can write the equation for our tangent line using the point slope form that we will have for point slope form, we'll have Y minus the Y value when X is equal to -2, which is 11 that we just calculated. Then this is going to be equal to our slope, which is -16, multiplying the quantity of X minus our value for X, which is -2. So, we can distribute that 16 on the right hand side of that expression. So we will have Y. -11 is equal to -16 X. Then we will have minus 32, since We will have 3 minus signs multiplied together, so it gives us an overall minus sign, then we'll just have 16 multiplied by 2, which is 32. So now, to solve for Y, we need to add 11 to both sides of this equation. And so we will have Y equal to -16 X minus 21. And so this is the equation for our tangent line. And if we compare our answer to our answer choices, the correct answer choice corresponds to answer choice D. So I hope that this explanation has been useful, and I'll see you in the next video.

{Use of Tech} Equations of tangent lines
Find an equation of the line tangent to the given curve at a.
y = −3x² + 2; a=1

Key Concepts

Tangent Line

Derivative

Point-Slope Form

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