Find the equation of the tangent line to f(x) = 11e^-0.5x at...

Find the equation of the tangent line to $f(x) = 11e^{-0.5x}$ at $x = 7$. NOTE: Enter the exact answer or round any calculated values to three decimal places. The tangent line equation is

Transcript

00:01 All right, so we're told that the function f of x is equal to 7 over x, so a hyperbolic function.

00:09 And we want to find the equation of the c -camp line from x equals 4 to x equals 7.

00:17 So if x is equal to 4, then f of x, we just plug in 4, is going to be 7 over 4.

00:24 And then at x equal to 7, f of 7 is equal to 3.

00:30 7 over 7, which is just 1.

00:34 So if we want to find the secan line, we take the y values, or f of 7 minus f of 4 for the slope, and then we divide by the x values.

00:47 So the rise over the run.

00:50 So we get 1 minus 7 over 4 divided by 3.

00:55 This is 4 over 4 minus 7 of 4.

00:57 So it's negative 3 over 4 divided by 3.

01:02 So that's just going to be negative 1 over 4.

01:05 Negative 1 4th is the slope of our secant line.

01:08 And then we can write it in slope form.

01:10 We know that y minus y1 is equal to m times x minus x1.

01:15 Let's use the point 7 comma 1 because there's, i don't want to deal with the fraction 7 over 4.

01:21 You could use either point and you'll get the same simplified equation.

01:25 So y minus 1, the y value here, is equal to the calculated slope negative 1 4th times x, minus the x value for that point is seven.

01:35 Then we can distribute the negative one -fourth.

01:38 I'm just getting it in slope intercept form.

01:41 So this is plus.

01:42 Oh, look at that...

Question

Find the equation of the tangent line to $f(x) = 11e^{-0.5x}$ at $x = 7$. NOTE: Enter the exact answer or round any calculated values to three decimal places. The tangent line equation is

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