Calculus II, MA 112
Quiz 8
NAME:
due Thursday, May 20, 2004
CM:
You will notice that each problem has a Maple part. Attach all your maple code and output
to the end of the quiz (please reduce the size of the graphs). Clearly write the problem
number on each problem so I can easily find them. You may need to use the view command
and adjust the limits when plotting.
(10 pts) 1. For the function f (x) = sin(x) with c = 0,
(a) find (by hand) the derivatives through f (5) , and then evaluate at c to determine P1 (x)
through P5 (x).
(b) Maple: compare the graphs of f (x), P1 (x), P3 (x), and P5 (x) on the interval [−2π, 2π].
(c) Maple: Use the taylor command to compare the graphs of f (x), P5 (x), P11 (x), and P15 (x).
(10 pts) 2. For the function f (x) = e−2x with c = 0,
(a) find (by hand) the derivatives through f (5) , and then evaluate at c to determine P1 (x)
through P5 (x).
(b) Maple: compare the graphs of f (x), P1 (x), P2 (x), and P3 (x) on the interval [−1, 8].
(c) Maple: Use the taylor command to compare the graphs of f (x), P5 (x), P10 (x), P15 (x),
and P16 (x).
(10 pts) 3. For the function f (x) =
1
with c = 0,
1−x
(a) find (by hand) the derivatives through f (5) , and then evaluate at c to determine P1 (x)
through P5 (x).
(b) Maple: compare the graphs of f (x), P1 (x), P3 (x), and P5 (x) on the interval [−2, 1].
(c) Maple: Use the taylor command to compare the graphs of f (x), P5 (x), P10 (x), P15 (x),
and P16 (x).
(10 pts) 4. For the function f (x) = arctan(x) with c = 0,
(a) find (by hand) the derivatives through f (3) , and then evaluate at c to determine P1 (x)
through P3 (x).
(b) Maple: compare the graphs of f (x), P1 (x), and P3 (x)on the interval [−2, 2].
(c) Maple: Use the taylor command to compare the graphs of f (x), P5 (x), P10 (x), P15 (x),
and P21 (x).
(10 pts) 5. For the function f (x) = ln(x) with c = 1,
(a) find (by hand) the derivatives through f (5) , and then evaluate at c to determine P1 (x)
through P5 (x).
(b) Maple: compare the graphs of f (x), P1 (x), P3 (x), and P5 (x) on the interval [0.1, 3].
(c) Maple: Use the taylor command to compare the graphs of f (x), P5 (x), P10 (x), P15 (x),
and P16 (x).
√
(10 pts) 6. For the function f (x) = x with c = 4,
(a) find (by hand) the derivatives through f (3) , and then evaluate at c to determine P1 (x)
through P3 (x).
(b) Maple: compare the graphs of f (x), P1 (x), P3 (x) on the interval [0, 10].
(c) Maple: Use the taylor command to compare the graphs of f (x), P5 (x), P10 (x), P17 (x),
and P31 (x).
P
(10 pts) 7. Graph ni=1 (−1)i+1 xi for n = 3, 10, and 20 (use Maple: use the sum command to define
the functions). You will discover that as n gets bigger, you will need to graph from −1 to 1.
This means that the radius of convergence of this particular power
P∞series isi+11. i You should
be able to guess from the shape of the graphs the function which i=1 (−1) x represents.
After making the correct guess, let x = 1 in both the infinite series and the function. This
will verify something that has been said in class.