2-4: Tangent Line Review &

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Presentation transcript:

2-4: Tangent Line Review & Linear Approximations

How to find the tangent line at x = a: Find f (a) unless you already know it (Plug a into the equation for y to find the point) Find (Plug a into to find the slope) Write the equation of the line

Find the equation of the line tangent to the function at x = 4 Remember that the slope of the tangent line is just the derivative at x = 4

Find the equation of the line tangent to the function at x = 4 But we want the whole equation, not just the slope But at x = 4, y = 2 so now we have the point (4, 2) Using point-slope, we get a final equation of

Approximate the square root of 5 without your calculator (remember, they haven’t always been around). But how? Let’s start with the function at x = 4 because we know what is First of all, what is the tangent line at x = 4? Now graph both the function and its tangent line…

Use the tangent line to the function at x = 4 to approximate Notice that for numbers close to 4, the tangent line is very close to the curve itself… So lets try plugging 5 into the tangent line equation and see what we get…

Use the tangent line to the function at x = 4 to approximate Notice how close the point on the line is to the curve… (5, 2.25) So let’s plug 5 into the tangent line to get…   (4, 2)  As it turns out… So in this case, our approximation will be a very good one as long as we use a number close to 4.

How to approximate values of f using the tangent line at x = a: Find the equation of the tangent line at x = a Plug the value of x for which you are trying to approximate f into the tangent line (not the original function)

Example 1 on page 95 (–1, 1 ) Use the tangent line of f at x = –1 to approximate f (–0.9) (–0.9, ?) So we are going to find the tangent line Then plug in –0.9 for x to the tangent line equation because that point on the line is so close to f (– 0.9)

(–1, 1 ) And since we have the point (–1, 1 ) , the equation of the tangent line is (–0.9, 0.7651) (–0.9, 0.7) or Now plug in –0.9 to approximate which is not far from the actual value:

So just to recap… How to approximate values of f using the tangent line at x = a: Find the equation of the tangent line at x = a Plug the value of x for which you are trying to approximate f into the tangent line (not the original function)