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Mathematics 2A: Single-Variable Calculus, Lecture Section C, Fall 2007
Homework
There will be no quiz based on this homework. Its material, however, will be included in the Midterms and in the Final Exam. All functions below have the domain consisting of all real numbers excluding those that would lead to division by zero or to taking a square root of a negative number. The abbreviation S&M refers to your textbook (Smith & Minton). Exercises marked “Optional” will not appear as mandatory on any quizzes or exams, but may appear as Extra Credit.
- Compute the following limits. If a limit does not exist, say so without proof. For improper limits, use the appropriate notation.
(a) S&M, section 1.5, exercise #1. (b) S&M, section 1.5, exercise #5. (c) S&M, section 1.5, exercise #7. (d) S&M, section 1.5, exercise #9. (e) S&M, section 1.5, exercise #13. (f) limx→π cot(x) (g) limx→π/ 4 tan( x (^2) +1x) (h) S&M, section 1.5, exercise #19. (i) limx→ 0 + |x x| (j) limx→ 0 − |x x| (k) limx→− 3 x
(^2) − 9 x+ (l) limx→ 1 x
(^3) − 1 x− 1 (m) limx→ 4 x
(^2) −x− 12 x− 4 (n) limx→ 0
√x+6−√x x (o) limx→ 0 √x+2x−√x
(p) limx→∞ (^) x (^3) +7^2 xx+5+
(q) limx→∞ 2 x
(^3) + − 4 x^3 − 7 x+ (r) limx→∞ 2 x
(^4) + x^3 − 7 x+ (s) limx→∞ 2 x
(^4) + − 4 x^3 − 7 x+ (t) limx→−∞ √^4 xx− (^2) +2^1
- Given two functions, f (x) and g(x), one must often know whether one of these “outgrows” the other as x increases without bound. One says that g is an upper bound for f as x → ∞ if there exists a constant C such that |f (x)| ≤ C|g(x)| for all x > some constant M (i.e., for all x in ]M, ∞[) If g is an upper bound of f , one writes f = O(g). The following theorem (which we will not prove) is often useful when determining whether one function is an upper bound for another: IF lim x→∞
f (x) g(x)
= L
(i.e., if the limit exists and equals L), THEN g is an upper bound of f as x → ∞. If L = 0, one says that g has a higher order of growth than f as x → ∞ and writes f = o(g). (The above definitions and theorem are completely analogous for x → a, where a is a real number; ]M, ∞[ is replaced by an appropriate neighborhood of a.)
(a) Show that g(x) = x^3 is an upper bound for f (x) = 126x^3 + 3047x^2 + x + 3 as x → ∞.
(b) Let n be a positive integer. Show that g(x) = xn^ is an upper bound for every polynomial of degree n or lower. (c) Determine which of the following two functions has a higher order of growth than the other:
f 1 (x) =
126 × 1037
x^5 + 42x^4 + 46789, f 2 (x) = 3x^5.^1
[Hint: To keep the huge coefficients from distracting you, let
a = 126 × 1037 , b = 42, c = 46789,
and rewrite f 1 in the form f 1 (x) = ax^5 + bx^4 + c.]
(d) (Optional) Suppose g(x) = f 1 (x) + f 2 (x), and f 1 = o(f 2 ) as x → ∞. Show that g = O(f 2 ) as x → ∞. [Hint: Consider the ratio g/f 2 .] Conclude that, when estimating the growth of a sum of terms, the slower-growing terms can be dropped. You encountered this basic principle when you did 2a and 2b.
- Suppose the functions f 1 (x), f 2 (x) are right-continuous at a. Deduce that
(a) the functions g 1 (x) = f 1 (x) + f 2 (x) and g 2 (x) = f 1 (x)f 2 (x) are also right-continuous at a. (b) if, in addition, f 2 (a) 6 = 0, then the function g 3 (x) = f 1 (x)/f 2 (x) is also right-continuous at a.
Same exercise for left-continuity and for continuity.
- Suppose the functions f 2 (x) is right-continuous at a, and the function f 1 (x) is right-continuous at x = f 2 (a). Deduce that the function g(x) = f 1 (f 2 (x)) is right-continuous at a. [Hint: Let y = f 2 (x) and rewrite
lim x→a+
f 1 (f 2 (x))
as lim y→f 2 (a)+
f 1 (y)
Use the right-continuity of f 1 to deduce that the latter limit equals f 1 (f 2 (a)).] Same exercise for left-continuity and for continuity.
- Suppose you are working on a problem in physics where a particle moves on a coordinate line. The function p(t) describes the position of the particle at time t (where t ≥ 0). For each of the following physical conditions, state the corresponding mathematical condition on p(t).
(a) The particle remains within distance d of a point a at all times. (b) The particle cannot instantly jump a positive distance. (c) The absolute value of the average velocity of the particle over any period of time [t 1 , t 2 ] does not exceed a constant C. (d) The particle eventually stops at position p 0.
- A park has a hiking trail of length L. The trail has no branches. The endpoints of the trail are called Start and Finish. A hiker starts along the trail and takes from 6 am to 6 pm to get from the Start to the point at distance 0. 8 L from the Start. She takes the rest of the evening to get to the Finish, sets up a tent, and settles down for the night. The next day, she begins hiking from the Finish toward the Start at 6 am, and by 6 pm reaches the point at distance 0. 3 L from the Start. Is there a time of day between 6 am and 6 pm at which the hiker is at the same distance from the Start on both days? Answer Yes or No, and justify your answer.
