Page 171 - Calculus Workbook For Dummies

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Chapter 8: Using Differentiation to Solve Your Problems

It’s a bit tricky to find the x-intercepts for this hairy function. You have to play around with the
window settings a bit. And don’t forget that your calculator will draw vertical asymptotes that
look like zeros of the function, but are not. To see the first zero, set xmin = –1, xmax = 10,
xscl = 1, ymin = –5, ymax = 25, and yscl = 5. To see the other two zeros, set xmin = 10, xmax = 30,
xscl = 1, ymin = –2, ymax = 10, and yscl = 1. The zeros are at roughly 6.11, 13.75, and 20.58.

4. Plug the zeros into the original function to obtain the y-coordinates. You get the following
points of normalcy: (6.11, 15.26), (13.75, 14.32), (20.58, 23.80).
2 2
5. Use the distance formula, D = _ x 2 - x 1 + _ y 2 - y 1i , to find the distance from our parched
i
adventurer to the three points of normalcy.
The distances are 14.77 miles to (6.11, 15.26), 17.07 miles to (13.75, 14.32), and 14.93 miles
to (20.58, 23.80). Using his trusty compass, he heads mostly south and a little east to
(6.11, 15.26). An added benefit of this route is that it’s in the direction of his camp.

s Estimate the 4th root of 17. The approximation is 2.03125.
1. Write a function based on the thing you’re trying to estimate: f x = 4 x
^ h
2. Find a “round” number near 17 where the 4th root is very easy to get: that’s 16, of course.
And you know 16 = 2. So (16, 2) is on f.
4
3. Determine the slope at your point.
f x = 4 x
^ h
1 - / 3 4
x =
f l ^ h x
4
1
f 16 =
l ^
h
32
4. Use the point-slope form of a line to write the equation of the tangent line at (16, 2).
1
2
y - = ^ x - 16h
32
5. Plug your number into the tangent line and you’ve got your approximation.
1
y = ^ 17 - 16 + 2
h
32
1
= 2 or . 2 03125
32
3
The exact answer is about 2.03054. Your estimate is only ⁄100 of 1 percent too big! Not too shabby.
Extra credit question (solve this or we may have to vote you off the island): No matter what 4th
root you estimate with linear approximation, your answer will be too big. Do you see why?
5
t Approximate .3 01 . The approximation is 247.05.
1. Write your function: g x = x 5
^ h
2. Find your round number. That’s 3, well duhh. So your point is (3, 243).
3. Find the slope at your point.
g x = 5 x 4
l ^ h
g 3 = 405
l ^ h
4. Tangent line equation.
y - y 1 = m x - x 1i
_
y - 243 405^ x - 3h
=
5. Get your approximation: y = 405^ . 3 01 - h 243 247 .05
=
3 +
1
Only ⁄100 of a percent off.