The italicized phrase before each sample problem indicates a similar problem in James Stewart. 2007. Essential Calculus: Early Transcendentals. Thomson Brooks/Cole: Belmont, CA.
Similar to 5.1.20:
5.1 Suppose Pm is the perimeter of a regular, m-sided polygon circumscribed about a circle with diameter d.
(a) Prove Pm = md tan (pi/m).
(b) Prove limm-->+infinity Pm = pi d.
Similar to 5.2.48:
5.2 Use the property below to estimate the value of the integral that follows it.
If L < f(x) < U for a < x < b, then
L(b - a) < integral from a to b of f(x) < U(b - a)
Integral from -1 to 3 of (3x^4 -5x^2 + 4x + 7) = ?
Similar to 5.3.46:
5.3 Between x = 0 and x = 1, find the area between the curves y = square root of x and y = cube root of x. Do this by solving each equation for x in terms of y, and then integrating with respect to y.
Similar to 5.3.64:
5.4 The area between x = 1 and x = a for y = 1/x is one thousandth the area between
x = 0 and x = b for y = e^x. Determine an equation for b in terms of a. What value of a yields b = 1?
Similar to 5.4.24:
5.5 Determine any intervals for which y is concave downward.
y = integral from 0 to x of [1 / (6 - 2u + 5u^2)] du
Similar to 5.4.32:
5.6 A waking nightmare company has just hired an evil monkey. Over time as the monkey sympathizes more with his target, the degree to which he scares and attacks his target will lessen by the measure a(t).
The evil monkey will also require more and more specialized foods, medical attention, matching contributions to his 401 K, etc., collectively measured by m(t).
(t is time measured in years. Both a(t) and m(t) are measured in thousands of dollars per year cubed.)
The company wants to determine the best time to offer the evil monkey early retirement.
(a) Assume the cost of the evil monkey through year t can be computed from the equation below:
C(t) = 7t^2 times the integral from 0 to t of [2a(s) + 3m(s)] with respect to s.
Prove that the critical values of C occur for the values of t where
C(t) = 3.5t^3 [2a(t) + 3m(t)].
For the rest of this problem, suppose a(t) = -2V + 2Vt for 0 < t < 2.42, a(t) = 0 for t > 2.42, and m(t) = +Vt^2.
(b) Find the time length T for all of the monkey's present value of evilness, V, to depreciate to zero. That is, when will V = integral from 0 to t of a(s) with respect to s?
(c) Find the absolute minimum of C for the interval (0,T].
(d) Sketch C and 3.5t^3 [2a(t) + 3m(t)] to confirm the part a conclusion for these a(t) and m(t) expressions.
Similar to 5.5.26:
5.7 Evaluate y.
y = integral of [tan (pi/x) / 3x^2] dx
Similar to 5.5.44:
5.8 Evaluate y.
y = integral from -pi/4 to pi/4 of [(3x^4 - 5x^2) tan x / (-7 + x^2)]
Similar to 5.5.46:
5.9 Determine y's value.
y = integral from 2 to 6 of [7x / cube root of (3 - 5x)] dx
Similar to 5.5.60:
5.10 Acme Phone Company built an assembly line to produce its triple screen smart phone. After t days, the line completes phones at this rate:
dx/dt = 12500 [1 - 276 / (277 + 13t)^4] phones/day
Determine the number of phones completed between the beginning of the 5th day and the end of the 21st day.
