I. Taylor Approximations.
A) (10) Use the error bound in Taylor approximations for
f(x) = exp(x), with a = 0, to find the degree of a
Taylor polynomial that could be used to approximate
sqrt(e) = e1/2 to
3 decimal place accuracy. (Some potentially useful information
Note that e < (1.7)2 = 2.89. So
e1/2 < 1.7.)
B) (10) Compute the Taylor approximation using the polynomial
of your degree from A, and check against a calculator value for
sqrt(e).
II. Taylor Series.
A) (15) Find the first 4 nonzero terms in the Taylor series
for sin(3t) with a = 0.
B) (5) For small values of t, the graph
y = sin(3t)/t is close to a parabola.
What is the equation of that parabola?
III. The velocity of a decelerating car is measured at each of the following times, yielding a table of values:
| t = (sec) | 1 | 3 | 5 | 7 | 9 | 11 |
|---|---|---|---|---|---|---|
| v(t) = velocity (ft/sec) | 10 | 9.5 | 8.1 | 5.8 | 4.0 | 1.2 |
A) (15) Give an estimate for the total distance traveled by the car
between t = 1 and t = 11 using a left-hand Riemann
sum for the velocity function.
B) (5) Is your result from A) less than or greater than
the actual total distance traveled, assuming v is always decreasing
between t = 1 and t = 11? Explain.
IV. Terminology.
A) (5) What is the definition of the definite integral
of a function f(x) over the interval [a,b]:
intab f(x) dx ?
B) (5) What does the Fundamental Theorem of Calculus
say about definite integrals?
V. Compute each of the following integrals:
A) (10) int14 x2 - 3 sqrt(x) + 4 dx
B) (10) int x2 e3xdx
C) (10) int sec2(2x) (tan(2x) + 4)1/3 dx