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Find the equation of the tangent to the curve π¦ equals negative two π₯ cubed plus eight π₯ squared minus 19 at π₯ equals two.
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Weβre looking for the equation of the tangent to this cubic curve.
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And we know that the slope of the tangent to the curve π¦ equals π of π₯ at π₯ equals π is the derivative π prime evaluated at π.
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So the slope of the tangent whose equation we want to find is the value of ππ¦ by ππ₯ at π₯ equals two.
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Hopefully, having found the slope of the tangent, weβll be well on our way to finding the equation of the tangent.
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This suggests that we should find ππ¦ by ππ₯, which is the derivative with respect to π₯ of negative two π₯ cubed plus eight π₯ squared minus 19.
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We can use the fact that the derivative of a sum or difference of functions is the sum or difference as appropriate of the derivatives of the functions to split this derivative into three.
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And we can evaluate these derivatives one by one starting with the derivative with respect to π₯ of negative two π₯ cubed.
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To find this derivative, we use the fact that the derivative with respect to π₯ of a power of π₯, π₯ to the π, is π times π₯ to the π minus one.
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And as the derivative of a number times a function is that number times the derivative of the function, the derivative with respect to π₯ of π times π₯ to the π is π times ππ₯ to the π minus one.
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So the derivative of negative two times π₯ to the three is negative two times three times π₯ to the three minus one, which is negative six π₯ squared.
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We can use the same rule to find the derivative of eight π₯ squared or eight π₯ to the two.
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This is 16π₯.
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And the derivative of the constant function 19 with respect to π₯ is just zero.
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This constant term doesnβt contribute to our derivative.
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And so ππ¦ by ππ₯ is negative six π₯ squared plus 16π₯.
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The slope of our tangent is ππ¦ by ππ₯ evaluated at π₯ equals two.
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So substituting two for π₯, we get negative six times two squared plus 16 times two, which is eight.
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Now that we found the slope of the tangent, letβs clear some room and find the equation of the tangent.
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We want to find the equation of the tangent line.
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And we know that the slope of this line is eight.
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But weβre going to need some other information to work out what the equation is.
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The tangentβs line touches or intersects the curve π¦ equals negative two π₯ cubed plus eight π₯ squared minus 19 when π₯ is two.
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So when π₯ is two, its π¦-coordinates must be the same as that of the curve.
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Hereβs a quick sketch to show why this fact is true in general.
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Applying this general fact to our example, we see that our tangent passes through the point two, negative two times two cubed plus eight times two squared minus 19.
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Here weβve just substituted two for π₯.
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Evaluating this, we find that the tangent passes through the point two, negative three.
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And now with these two pieces of information, we have enough information to find the equation of the tangent.
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We can use the point slope form of the equation of a line and substitute in.
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The slope π is eight.
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And π₯ naught and π¦ naught are two and negative three, respectively, because the tangent passes through the point two, negative three.
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All we have to do now is simplify.
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On the left-hand side, minus negative three becomes plus three.
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And on the right-hand side, we expand to get eight π₯ minus 16.
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Rearranging the equation so that all the terms fall on the left-hand side, we find that the equation of the tangent to the curve π¦ equals negative two π₯ cubed plus eight π₯ squared minus 19 at π₯ equals two in standard form is π¦ minus eight π₯ plus 19 equals zero.