72–76. Tangent lines Find an equation of the line tangent to e...

Question

72–76. Tangent lines Find an equation of the line tangent to each of the following curves at the given point.

y = 3x³+ sin x; (0, 0)

Accepted Answer

Hello there. Today we're going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Determine the equation of the tangent line to the function Y equals 4 x 3 minus 2 multiplied by sin of x at the point 0.0 in parentheses. Awesome. So it appears we're ultimately trying to figure out what the equation of the tangent line is to this function of Y at the specific point of 0.0. Awesome. So with that said, let's read off our multiple choice answers to see what our final answer might be, noting that they're all state Y equals some expression. So A is -2 X, B is 2 X, C is negative X, and D is X. Awesome. So first off, we need to recall a note that in order to find the tangent line to our equation of Y equals 4 X 3 minus 2 multiplied by sin of x, at the point 0.0 in parentheses, we need to first figure out the derivative. So in order to find the derivative of, so we're trying to find the derivative of Y is equal to 4 x 3 minus 2 multiplied by the sine of X. So, in order to perform this derivative, we need to recall and use the power rule to differentiate this 4 X to the power of 3. Which, as we should recall, this 4 X to the power of 3, when we take the derivative, specifically the first derivative, it will give us 12 x to the power of 2. And we need to note and recall that the derivative of -2, multiplied by sin of X, this will be Our derivative of -2 multiplied by sin of X will be -2 multiplied by cosine of X. Fantastic. So therefore, when we combine the results, that will mean that our first derivative of the function of Y, which will denote as Y, which once again this prime indicates the first derivative. So Y will be equal to 12 x 2 minus 2 multiplied by cosine of X. Fantastic. And now our next step is we need to evaluate the derivative at X equals 0. So when the evaluator derivative at X equals 0, we will find that Y 0 is equal to 12 multiplied by 0 squad because once again we're plugging in a value of 0 in for X. -2 multiplied by cosine of 0. And when we plug that into our calculator, we will find that it's equal to -2. So now our next step is we need to recall and use the point slope form and plug in our known variables. So Y minus 0, because we're told that our coordinate point is 0.0, which, as we should recall, is in the XY coordinate layout, so X. Y. So Y minus 0 is equal to -2 multiplied by X minus 0 because we determined. When we evaluate the derivative at X equals 0, we got -2, and this -2 is our value for M, the slope. So when we rearrange our point slope form equation to isolate and solve for Y, we will find that the equation of the tangent line is equal to. Y is equal to -2 X. And that's it. That is our final answer. Hooray, we did it. So I skipped a few steps, but essentially you need to take your point slope form and you're rearranging it to isolate and solve for Y on one side. So when you do that, you will find that Y is equal to -2 X. And this corresponds with multiple choice answer letter A, which once again our final answer is Y equals -2 X. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

72–76. Tangent lines Find an equation of the line tangent to each of the following curves at the given point.

y = 3x³+ sin x; (0, 0)

Key Concepts

Tangent Line

Derivative

Point-Slope Form

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