10. Derivatives and Tangent Lines
b. Tangent Lines
The primary application of a derivative is to find the equation of the tangent line at a point, \(x=a\), on the graph of a function, \(y=f(x)\). The most general line has the standard equation: \[ Ax+By=C. \] The line is vertical if \(B=0\) and non-vertical if \(B\ne0\). Assuming it is not vertical, we can solve for \(y\) and put the equation into slope-intercept form: \[ y=mx+b \qquad (1) \] where \(m\) is the slope and \(b\) is the \(y\)-intercept.
Now suppose we want to find the equation of the line tangent to \(y=f(x)\) at \(x=a\). We know the slope is \(m=f'(a)\). So equation \((1)\) becomes: \[ y=f'(a)x+b \qquad (2) \] We know the line passes through the point \((x,y)=(a,f(a))\). So equation \((2)\) tell us: \[ f(a)=f'(a)a+b \] or \[ b=f(a)-f'(a)a \] Using this formula for \(b\), equation \((2)\) becomes: \[\begin{aligned} y&=f'(a)x+f(a)-f'(a)a\\ &=f(a)+f'(a)(x-a) \end{aligned}\] which is the equation for the tangent line. We define the formula on the right to be the tangent function: \[ f_{\tan}(x)=f(a)+f'(a)(x-a) \] so that the equation of the tangent line is \(y=f_{\tan}(x)\).
The equation of the tangent line to the graph of the function \(y=f(x)\) at \(x=a\) is: \[ y=f_{\tan}(x) \equiv f(a)+f'(a)(x-a). \] In differential notation, this is: \[ y=f_{\tan}(x) \equiv f(a)+\left.\dfrac{df}{dx}\right|_{x=a}(x-a). \]
Memorize this!
Compute the tangent line to \(y=6x^2-x^3\) at \(x=3\). Then find its \(y\)-intercept. Its slope was found in an exercise on the previous page.
The solution is straightforward: find the required information and plug into the formula for the equation of the tangent line. We identify \(f(x)=6x^2-x^3\) and \(a=3\). Then: \[ f(3)=54-27=27 \] and from the previous page: \[ f'(3)=9 \] So the equation of the tangent line is: \[\begin{aligned} y&=f(a)+f'(a)(x-a) \\ &=f(3)+f'(3)(x-3) \\ &=27+9(x-3) \\ &=9x \end{aligned}\] We identify the \(y\)-intercept as \(b=0\).
Compute the tangent line to \(y=\dfrac{3}{x}\) at \(x=6\). Then find its \(y\)-intercept. Its derivative was found in an exercise on a previous page.
Evaluate the function and find its derivative at \(x=6\). Then use the formula: \[ f_{\tan}(x)=f(6)+f'(6)(x-6) \]
\(y=-\,\dfrac{1}{12}x+1\)
The \(y\)-intercept is \(b=1\).
We identify \(f(x)=\dfrac{3}{x}\) and \(a=6\). Then: \[ f(6)=\dfrac{1}{2} \] The derivative was found on a previous page to be: \[ f'(6)=-\,\dfrac{1}{12} \] So the equation of the tangent line is: \[\begin{aligned} y&=f(a)+f'(a)(x-a) \\ &=f(6)+f'(6)(x-6) \\ &=\dfrac{1}{2}-\dfrac{1}{12}(x-6) \\ &=-\,\dfrac{1}{12}x+1 \end{aligned}\] We identify the \(y\)-intercept as \(b=1\).
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