Write up the following problems and hand them in:
section 2.1, page 117: 10, 18, 28, 30, 32, 38, 50, 52
section 2.2, page 129: 14, 24, 40, 42, 50, 52a, 54a
section 2.3, page 139: 12, 18, 28, 32
section 2.4, page 152: 18, 34, 42, 44, 54, 69-72 (use maple).
Do the following problems in maple and hand in your
print-outs. Go through the
sample worksheet for help.
Problems with Maple? Here are the
commands I used.
I've been referring to the tangent lines through the point (p,f(p)) as
an "approximation" of the graph of f near the point (p,f(p)). In the
following problems, you will get a better feeling for what I meant by this.
-
Consider f(x) = 1/x at the point 10. Find L, the tangent line through
(10,f(10)).
First plot both f and L on the same plot, plotting it
from x=10-.1 to x=10+.1. Then plot the difference f-L, plotting it
from x=10-.1 to x=10+.1.
Now plot both f and L on the same plot, plotting it
from x=10-.01 to x=10+.01. Then plot the difference f-L, plotting it
from x=10-.01 to x=10+.01.
Finally plot both f and L on the same plot, plotting it
from x=10-.001 to x=10+.001. Then plot the difference f-L, plotting it
from x=10-.001 to x=10+.001.
Discuss what you observed as you "zoomed in".
-
Consider f(x) = 1/x at the point 1. Find L, the tangent line through
(1,f(1)).
First plot both f and L on the same plot, plotting it
from x=1-.1 to x=1+.1. Then plot the difference f-L, plotting it
from x=1-.1 to x=1+.1.
Now plot both f and L on the same plot, plotting it
from x=1-.01 to x=1+.01. Then plot the difference f-L, plotting it
from x=1-.01 to x=1+.01.
Finally plot both f and L on the same plot, plotting it
from x=1-.001 to x=1+.001. Then plot the difference f-L, plotting it
from x=1-.001 to x=1+.001.
Discuss what you observed as you "zoomed in".
-
Consider f(x) = 1/x at the point .2. Find L, the tangent line through
(.2,f(.2)).
First plot both f and L on the same plot, plotting it
from x=.2-.1 to x=.2+.1. Then plot the difference f-L, plotting it
from x=.2-.1 to x=.2+.1.
Now plot both f and L on the same plot, plotting it
from x=.2-.01 to x=.2+.01. Then plot the difference f-L, plotting it
from x=.2-.01 to x=.2+.01.
Finally plot both f and L on the same plot, plotting it
from x=.2-.001 to x=.2+.001. Then plot the difference f-L, plotting it
from x=.2-.001 to x=.2+.001.
Discuss what you observed as you "zoomed in".
-
Consider f(x) = 1/x at the point -.2. Find L, the tangent line through
(-.2,f(-.2)).
First plot both f and L on the same plot, plotting it
from x=-.2-.1 to x=-.2+.1. Then plot the difference f-L, plotting it
from x=-.2-.1 to x=-.2+.1.
Now plot both f and L on the same plot, plotting it
from x=-.2-.01 to x=-.2+.01. Then plot the difference f-L, plotting it
from x=-.2-.01 to x=-.2+.01.
Finally plot both f and L on the same plot, plotting it
from x=-.2-.001 to x=-.2+.001. Then plot the difference f-L, plotting it
from x=-.2-.001 to x=-.2+.001.
Discuss what you observed as you "zoomed in".
-
Now that you've studied f(x) = 1/x near the points, x = 10, 1, .2, and
-.2, discuss the relations and trends you observed when studying
x=10, 1, and .2. Also discuss the relation between what you observed
for x=.2 and x=-.2
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