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05:02[Technology Exercise] Let f(t) = 1/t for t≠0.
a. Find the average rate of change of f with respect to t over the intervals (i) from t=2 to t=3, and (ii) from t=2 to t=T.
b. Make a table of values of the average rate of change of f with respect to t over the interval [2,T], for some values of T approaching 2, say T = 2.1, 2.01, 2.001, 2.0001, 2.00001, and 2.000001.
c. What does your table indicate is the rate of change of f with respect to t at t=2?
The accompanying graph shows the total distance s traveled by a bicyclist after t hours.
b. Estimate the bicyclist’s instantaneous speed at the times t=1/2, t=2, and t=3.
[Technology Exercise] Roots
Let ƒ(𝓍) = 𝓍³ ―𝓍― 1.
b. Solve the equation ƒ(𝓍) = 0 graphically with an error of magnitude at most 10⁻⁸ .
Finding Limits
In Exercises 9–24, find the limit or explain why it does not exist.
lim x →π cos² (x― tan x)
Finding Deltas Algebraically
Each of Exercises 15–30 gives a function f(x) and numbers L, c, and ε>0. In each case, find the largest open interval about c on which the inequality |f(x)−L| <ε holds. Then give a value for δ>0 such that for all x satisfying 0 < |x−c| < δ, the inequality |f(x)−L| < ε holds.
f(x) = mx, m > 0, L = 2m, c = 2, ε = 0.03
[Technology Exercise] Roots
Let ƒ(𝓍) = 𝓍³ ―𝓍― 1.
c. It can be shown that the exact value of the solution in part (b) is
(1/2 + √69/18)¹/³ + (1/2 ― √69/18)¹/³
Evaluate this exact answer and compare it with the value you found in part (b).
