quiz
Problems 1. (a) State the definition of the derivative. (b) What is the graphical interpretation of the derivative? (c) What is the equation of the line tangent to y = g(x) at x = x1? 2. Compute the following limits. If the limit does not exist, analyze its one-sided limits, and determine if the limit approaches ∞, −∞, or neither. (a) lim x→−5 −3x2 − 1 2x+ 5 (b) lim θ→ 1 2 πθ (c) lim x→3 x2 − 9 |x− 3| (d) lim x→2 10− 10 √ 3− x x− 2 (e) lim h→0 √√√√ 12 − 1√4 + h h (f) lim x→−3 1 9 + 1 x2 x+ 3 (g) lim x→2 (x+ 1)3 − 27 x− 2 (h) lim x→3 (2x+ |x− 3|) (i) lim x→0 ( 1 x − 1 |x| ) (j) lim x→7 exp ( x2 − 5x− 14 x2 − 11x+ 28 ) 3. Find the domain of the following functions. Analyze the limits at points outside the domain, splitting into one-sided limits as necessary. (a) g1(x) = x2 + 3x− 10 x2 + 2x− 15 (b) g2(x) = x2 − x x4 − x3 4. True / False. Justify your answerr: if the answer is false, provide a counter example (or explain why its false); otherwise, write a couple words or a sentence describing why its true). (a) If f(s) = f(t) then s = t. (b) If f and g are functions then f ◦ g = g ◦ f . 1 (c) lim x→4 ( 2x x− 4 − 8 x− 4 ) = lim x→4 ( 2x x− 4 ) − lim x→4 ( 8 x− 4 ) . (d) If lim x→5 f(x) = 2 and lim x→5 g(x) = 0, then lim x→5 f(x) g(x) does not exist. (e) If lim x→5 f(x) = 0 and lim x→5 g(x) = 0, then lim x→5 f(x) g(x) does not exist. (f) If g(1) = −1 and g(2) = 5 then there exists a number c between 1 and 2 such that g(c) = 0. (g) If 1 ≤ f(x) ≤ x2 + 2x+ 2 for all x near −1, then lim x→−1 f(x) = 1. (h) If the line x = 1 is a vertical asymptote of y = f(x), then f(1) does not exist. 5. Determine the constant c that makes f continuous on (−∞,∞). f(x) = { c2 + sin (xπ) if x < 2="" cx2="" −="" 4="" if="" x="" ≥="" 2="" .="" 6.="" find="" a="" value="" of="" b="" so="" that="" l="lim" x→−2="" 3x2="" +="" bx+="" b+="" 3="" x2="" +="" x−="" 2="" exists.="" then="" find="" l.="" 7.="" consider="" the="" function="" f(x)="9−" 4e2x="" e2x="" −="" 2ex="" −="" 3="" (a)="" find="" the="" equation="" of="" all="" horizontal="" asymptotes.="" (b)="" find="" the="" equation="" of="" all="" vertical="" asymptotes.="" (c)="" analyze="" the="" one-sided="" limits="" at="" the="" vertical="" asymptotes.="" 8.="" in="" the="" theory="" of="" relativity,="" the="" lorentz="" contraction="" formula="" l="L0" √="" 1−="" v2/c2="" expresses="" the="" length="" l="" of="" an="" object="" as="" a="" function="" of="" its="" velocity="" v="" with="" respect="" to="" an="" observer,="" where="" l0="" is="" the="" length="" of="" the="" object="" at="" rest="" and="" c="" is="" the="" speed="" of="" light.="" find="" lim="" v→c−="" l="" and="" interpret="" the="" result.="" why="" is="" a="" left-hand="" limit="" necessary?="" 9.="" evaluate="" lim="" x→2="" √="" 6−="" x−="" 2√="" 3−="" x−="" 1="" 10.="" (precalculus="" review.)="" solve="" for="" x.="" 2="" (a)="" lnx+="" ln(x−="" 1)="1" (b)="" 2x−5="43x+7" 11.="" (precalculus="" review.)="" find="" the="" inverse="" of="" the="" following="" invertible="" functions:="" (a)="" f(x)="2" +="" x="" 1−="" x="" .="" (b)="" g(x)="3ex" 1="" +="" 2ex="" 12.="" prove="" lim="" x→−∞="" ex="" cos(x)="0." 13.="" prove="" log="" x="x−" 3="" has="" a="" solution.="" hint:="" recall="" log(x)="log1" 0(x).="" 14.="" prove="" that="" f(x)="" is="" not="" continuous="" at="" x="−3." then,="" determine="" whether="" f(x)="" is="" continuous="" from="" the="" left,="" continuous="" from="" the="" right,="" or="" neither="" at="" x="−3." f(x)="" x2="" +="" 4x+="" 3="" x+="" 3="" if="" x="">< −3="" −2="" if="" x="−3" x+="" 5="" if="" x=""> −3 15. Suppose the displacement function of a particle is given by s(t) = √ t7 for t > 0. (a) Using the limit definition of the derivative, compute the velocity function. (b) Compute the acceleration function using the definition of the derivative. 16. For which a, b ∈ R is the function f(x) = √ 1− x− 1 ax if x ∈ (0, 1] 1 if x = 0 bx4 + bx x2 + x if x ∈ (−1, 0) continuous on (−1, 1]? 17. Compute the following limits. (a) lim x→∞ ( √ x2 + ax− √ x2 + bx) (b) lim x→−∞ ( √ x2 + ax− √ x2 + bx) 3 18. The graph of y = f(x) is given. On the same set of axes, sketch f ′(x). 19. Suppose f(x) = x3 + x− 8. Find the equation of the tangent line to f(x) at x = −1. 20. State f(x) = |x− 3| x− 3 as a piecewise function. Then show that lim x→3 f(x) does not exist by a) plotting f(x) and b) analytically. 21. Let f(x) = 1 x2 − x . (a) Calculate f ′(2) using the definition of the derivative. (b) Find the tangent line to the curve y = f(x) at the point x = 2. 22. Determine the value of c so that g(x) is continuous g(x) = x2 − 9 x− 3 if x < 3="" cx−="" 5="" if="" x="" ≥="" 3="" 23.="" the="" gravitational="" force="" exerted="" by="" the="" earth="" on="" a="" unit="" mass="" at="" a="" distance="" r="" from="" the="" centre="" of="" the="" planet="" is="" f="" (r)="" gmr="" r3="" if="" r="">< r gm r2 if r ≥ r where m is the mass of the earth, r is its radius, and g is the gravitational constant. is f a continuous function of r? 24. challenging problem. find a formula for a function f that satisfies the following conditions: (a) lim x→±∞ f(x) = 0 (b) lim x→0 f(x) = −∞ 4 (c) f(2) = 0 (d) lim x→3− f(x) =∞ (e) lim x→3+ f(x) = −∞ 5 solutions 1 (a) f ′(x) = lim h→0 f(x+ h)− f(x) h (b) slope of tangent line (c) y = g′(x1)(x− x1) + g(x1) 2 (a) -74/5 (b) √ π (c) lhl -6 rhl 6 (d) 5 (e) 1/4 (f) lhl −∞ rhl ∞ (g) 19 (h) 6 (i) lhl −∞ rhl 0 (j) e3 3 3 4 (a,b,c,e,f,h) are false; (d,g) are true. remark: i’ve double-checked these and i am confident there are no typos. 5 c = 2 6 b = 15 l = −1 7 (a) y = −7 (b) x = ln 3 (c) lhl =∞ rhl = −∞ vertical asymptote is x = ln 2; horizontal asymptote is y = −3 8 l→ 0+ 9 hint: rationalize relentlessly. answer: 1/2. 10 11 (a) f−1(x) = x− 2 1 + 3x (b) g−1(x) = ln ( x 3− 2x ) 12 hint: use squeeze theorem. 13 hint: use the intermediate value theorem. 14 continuous from the left. 15 16 a = −1/2, b = 1 17 (a) a− b 2 (b) b− a 2 18 19 20 (b) hint: split into left- and right- hand limits. 21 y = 2− 3x/4 22 c = 11/3 23 hint: state the definition of continuity for f (r) at r = r. yes, f (r) is continuous. 24 one solution is f(x) = −(x− 2)/x2(x− 3) 6 r="" gm="" r2="" if="" r="" ≥="" r="" where="" m="" is="" the="" mass="" of="" the="" earth,="" r="" is="" its="" radius,="" and="" g="" is="" the="" gravitational="" constant.="" is="" f="" a="" continuous="" function="" of="" r?="" 24.="" challenging="" problem.="" find="" a="" formula="" for="" a="" function="" f="" that="" satisfies="" the="" following="" conditions:="" (a)="" lim="" x→±∞="" f(x)="0" (b)="" lim="" x→0="" f(x)="−∞" 4="" (c)="" f(2)="0" (d)="" lim="" x→3−="" f(x)="∞" (e)="" lim="" x→3+="" f(x)="−∞" 5="" solutions="" 1="" (a)="" f="" ′(x)="lim" h→0="" f(x+="" h)−="" f(x)="" h="" (b)="" slope="" of="" tangent="" line="" (c)="" y="g′(x1)(x−" x1)="" +="" g(x1)="" 2="" (a)="" -74/5="" (b)="" √="" π="" (c)="" lhl="" -6="" rhl="" 6="" (d)="" 5="" (e)="" 1/4="" (f)="" lhl="" −∞="" rhl="" ∞="" (g)="" 19="" (h)="" 6="" (i)="" lhl="" −∞="" rhl="" 0="" (j)="" e3="" 3="" 3="" 4="" (a,b,c,e,f,h)="" are="" false;="" (d,g)="" are="" true.="" remark:="" i’ve="" double-checked="" these="" and="" i="" am="" confident="" there="" are="" no="" typos.="" 5="" c="2" 6="" b="15" l="−1" 7="" (a)="" y="−7" (b)="" x="ln" 3="" (c)="" lhl="∞" rhl="−∞" vertical="" asymptote="" is="" x="ln" 2;="" horizontal="" asymptote="" is="" y="−3" 8="" l→="" 0+="" 9="" hint:="" rationalize="" relentlessly.="" answer:="" 1/2.="" 10="" 11="" (a)="" f−1(x)="x−" 2="" 1="" +="" 3x="" (b)="" g−1(x)="ln" (="" x="" 3−="" 2x="" )="" 12="" hint:="" use="" squeeze="" theorem.="" 13="" hint:="" use="" the="" intermediate="" value="" theorem.="" 14="" continuous="" from="" the="" left.="" 15="" 16="" a="−1/2," b="1" 17="" (a)="" a−="" b="" 2="" (b)="" b−="" a="" 2="" 18="" 19="" 20="" (b)="" hint:="" split="" into="" left-="" and="" right-="" hand="" limits.="" 21="" y="2−" 3x/4="" 22="" c="11/3" 23="" hint:="" state="" the="" definition="" of="" continuity="" for="" f="" (r)="" at="" r="R." yes,="" f="" (r)="" is="" continuous.="" 24="" one="" solution="" is="" f(x)="−(x−" 2)/x2(x−="" 3)="">