ANSWERS / MATH 75 = CALCULUS I HOMEWORK #4, DUE Wed 2 OCT 2002 ----------------WARREN D. SMITH-------------------------- 1. Do plotting problems 1,2,3 and 5 on page 83 of the HH book. 2. Problem 14 page 99 of HH book. F(x) = 1+cos(x pi/2) = 0 if |x| <= 2 0 if |x| > 2 Yes continuous and yes differentiable at x=2. Because cos(2pi/2) = cos(pi) = -1, so 1+cos(x pi/2) = 0 when x=2, which joins CONTINUOUSLY to the SAME value 0 for x>2. No jump in value. It is DIFFERENTIABLE since cos(x pi/2) has a MINIMUM value of -1 at x=2, i.e. is of slope=0 here, which is the same as the slope of the CONSTANT (horizontal line) 0. No jump in slope. Same limiting slope no matter from which side of x=2 you come. 3. Problem 16 page 100 of HH book. After doing this: What is a formula, in terms of x, for the derivative of the square root of x (assuming x is positive)? ANSWER: 1/2 1/2 (x + h) - x h 1 ----------------- = ---------------------- = ---------------- h 1/2 1/2 1/2 1/2 h [(x + h) + x ] (x + h) + x by multiplying both top and bottom by 1/2 1/2 (x + h) + x then expanding out the top by the distributive law using 2 2 (A-B)(A+B) = A - B then cancelling the +x with -x, then cancel the h on top and bottom. If we take the limit as h-->0 then this becomes 1 1 1/2 ---------- = ------ = F'(x) if F(x) = x. 1/2 1/2 1/2 x + x 2 x. PROBLEM 4. Derive a formula (in terms of trig functions) for the derivative of tan(x). Hints: Use the tan addition formula tan(a) + tan(b) tan(a + b) = ----------------- 1 - tan(a) tan(b) to help write a formula for the slope (=rise/run) of the tan(x) function based on the two x-values x and x+h, which you can simplify (for example to a form only involving sin's and cos's). Then take the limit as h-->0. Hint You may use the fact that lim sin(k)/k = 1. k-->0 Simplify the final answer to get a nice clean formula for the derivative of tan(x) in terms of x only. (Now: Can you think of some sanity checks to see if your formula is sane?) ANSWER: tan(x+h) - tan(x) tan'(x) = lim ----------------- h-->0 h tan(x) + tan(h) ---------------- - tan(x) 1 - tan(x)tan(h) = lim ------------------------- (used addition formula for tan) h-->0 h = lim tan(x) + tan(h) - tan(x) [ 1 - tan(x) tan(h) ] (put over h-->0 ---------------------------------------------- common denom.) [ 1 - tan(x) tan(h) ] h 2 = lim tan(h) + tan(x) tan(h) (canceled +tan(x) with h-->0 ------------------------- -tan(x) on top; noted -- = +) [ 1 - tan(x) tan(h) ] h now the tan(x)tan(h) on the bottom is tiny compared to 1 so neglect it 2 = lim tan(h) + tan(x) tan(h) h-->0 ----------------------- h 2 tan(h) = lim [ 1 + tan(x) ] ------ h-->0 h Now we can use sin(h)/h --> 1 when h-->0 from the hint, and cos(0)=1 so tan(h)/h = sin(h)/[h cos(h)] --> 1/1 = 1. Another way to see this is draw a right triangle with horizontal leg 1, angle h, and vertical leg (therefore) tan(h). The vertical leg is, in the limit, very close to the length of the circular arc of radius 1 and angle h, (the earth locally looks flat! Measuring it with a straight ruler gives same answer as with flexible ruler in limit of small lengths being measured!) so tan(h)/h --> 1 as h-->0. So... 2 2 2 cos(x) + sin(x) 1 tan'(x) = 1 + tan(x) = ------------------- = ------- 2 2 cos(x) cos(x) SANITY CHECK: tan(x) increases from 0 to infinity as x goes from 0 to pi/2. So the deriviative should be POSITIVE (since increasing) and it should go infinite at x=pi/2 (yes: cos(pi/2)=0) and it should initially be 1 at x=0 since tan(x)=x+small discrepancy for x small (by geometric argument about nearly-flat earth). Yes, we pass these sanity checks! PROBLEM 5. Consider your height H(t) as a function of time t for the first 18 years of your life. The derivative H'(t) is your "growth rate". What was your approximate average growth rate during these years, in units of nanometers per second? (A nanometer is 10 to the power -9 meters. About 10 atoms lined up next to each other make 1 nanometer.) Now assume the American and European continents have drifted apart (they used to be touching) during the last 200 million years. What average drift rate (derivative of their distance D(t) apart) was that (in the same units - nanometers per second - please)? Which speed is faster - you growing or the continents separating? ANSWER: The number of seconds in a year is 1 year = 8*365*24*60*60 seconds = 31536000 seconds. I grew about 1.5 meters from age 0 to age 18. 9 1.5 meters 1.5 X 10 nanometers nanometer So H'(t) = ---------- = --------------------- = 2.6 --------- average 18 years 18 X 31536000 seconds second Continents drifted from touching to about 5000 km apart in 200 million years 3 3 9 5000 km 5 X 10 X 10 X 10 nanometers nanometer speed = ----------------- = ---------------------------- = 0.8 ---------- 200 million years 6 second 200 X 31536000 X 10 seconds so people grow faster than continents drift! But not by much!