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Math 1151 Test 11: Derivatives and Inverse Functions Exam Guide

School
Lakehead University**We aren't endorsed by this school
Course
MATH 1151
Subject
Mathematics
Date
Dec 11, 2024
Pages
10
Uploaded by MinisterWater45010
Lakehead University Department of Mathematical Sciences MATH 1151 Test 11 November 15, 2024 Location: RB 1042 Time: 12:30 pm - 1:20 pm INSTRUCTIONS: The duration of the exam is 50 minutes. This exam contains 5 pages (including this cover page). Check to see if any pages are missing. Books, notes, calculators, cellphones or other aids are not permitted. Cell phones must be completely turned off and placed in your bag in the front of the room. e Provide your answers in the space provided . If you need more space, use the back of the pages; clearly indicate when you have done this. ¢ Organize your work, in a reasonably neat and logical way, in the space provided. Work scattered all over the page without a clear ordering will receive very little credit. o Justify your answers. Unsupported answers will not receive credit. Page | Points | Score 2 14 CAME SOLUTToWS T . 4 7 STUDENT NUMBER: 5 8 Total: 35
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MATH 1151 Test II - Page 2 of November 15, 2024 SHORT ANSWER: Provide your answers in the space given. 1. Compute the derivative of each of the following functions. Simplification is not required. (a) (2 points) f(z) = In(z® +4z) £'w0= St Bl xS+4x . _dsec(t) +1%° (b) (4 points) g(t) = —53 o g10= fsectpmls) + 34 %) @€st)- ( p0043) (45t 1Y) (23rat)” (c) (4 points) G(z) = (28 +3* ~ Tz 3 arcsin(z) G N (%K +3M3) - ~21¢) avesii) & ( qrgx—Yx) [-x* (d) (4 points) H (z) = 64; j‘lgl e 4lox-1) ~ blx+q) 2 = (ex-1)"
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MATH 1151 Test II - Page 3 of & November 15, 2024 2. (6 points) Find —Z—i for the equation x4y3 + 2% = 3sin(y ) Aot o (x"z ) ’A— (38!\%3) % 5 )+ 2) > 2 (35) APyt Ayt + B = Busl AT BT By o 3ot ey} A o (’;’,x‘yl ”Scos(%)\)‘%g N P ‘f/x?y 3 i'/(j_ _ ‘__%/)(7__{_‘\)(3‘.\'3 M 39{‘31 “5@3(%
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MATH 1151 Test II - Page 4 of 5 November 15, 2024 3. Let f(z) =1+ 3z +2. (a) (2 pomts) Show that f is a one-to-one function. (b) (3 points) Derive the formula for the inverse function f™(z). (c) (2 points) Find the equation of the tangent line to the curve y = f~ Yz) at ¢ = 3. @ ket £00=fxy) . 3! = | 44 s /13x, 1 v[ y S xr L=kt Xy >3 X =K A";\UL ’HXO';‘((XL) MA’)\W W(’X‘vl 7L£<fl’(- SR x-\= E\FT Q“')L = 3qrl 2y - x--2 ) %'\)\(‘3); q/,?)_(x/o <‘F—\)(3> - (%:%‘:—/'L : 7}3 P’{’ (7)’ /5) B4 \) vy > & \/ w w 1 ~/ \ > 3 3 Q (@ Wl \ \N \\ WYL 3 9
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MATH 1151 Test I - Page 5 of 5 November 15, 2024 4. (4 points) Compute the derivative of f (z) = z%. (6t} = b (3] Do (£61) = 222 nlx) ;\‘3‘;— LM%\xD:&% (2x b«(x)) /?\(-X) - QJM(’Q'\' 92X )),(_ £ $60= Unlx) + L £ ff )X [zwm*zl 5. (4 points) A circular pond’s area is increasing at a rate of 5 cm?/s. At the moment when the radius is 6 cm, how fast is the radius of the pond expanding? \) % > SOMI/S Fano if_ i, r=6 on AT e I i oMY L 2rdr A v S = 2x(b) i At 5 = 2w oX redlVs B mgentiny oA 4l of v wfs
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Lakehead University Department of Mathematical Sciences MATH 1151 Test 11 November 15, 2024 Location: RB 1042 Time: 12:30 pm - 1:20 pm INSTRUCTIONS: The duration of the exam is 50 minutes. This exam contains 5 pages (including this cover page). Check to see if any pages are missing. Books, notes, calculators, cellphones or other aids are not permitted. Cell phones must be completely turned off and placed in your bag in the front of the room. e Provide your answers in the space provided . If you need more space, use the back of the pages; clearly indicate when you have done this. e Organize your work, in a reasonably neat and logical way, in the space provided. Work scattered all over the page without a clear ordering will receive very little credit. ¢ Justify your answers. Unsupported answers will not receive credit. Page | Points | Score 2 14 NAME: DLUTToNS s | 6 4 7 STUDENT NUMBER: 5 8 Total: 35
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MATH 1151 Test 11 - Page 2 of 5 November 15, 2024 SHORT ANSWER: Provide your answers in the space provided. 1. Compute the derivative of each of the following functions. Simplification is not required. (a) (2 points) f(z)= In(z” + 3z) FR=Te3 T3 . _5 tan(t) + 1/ (b) (4 points) g(t) EToan () - Q—ésec wr 1) 3% 5t ) (bt ) (s'x\me()»ft‘@) 3 (3t3st)” (c) (4 points) G(z) = (z° +5° — 622) arctan(z) G'b= (94 5n) - 12x) orchunld x4 6 L+ x*- (d) (4 points) H(x) = A/ 22 +;\) - /\’3 61‘ x / Hx) = | 3(5x-a) - 5(3x7) 2,31 (Bx-” ~—
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MATH 1151 Test 11 - Page 3 of § November 15, 2024 2. (6 points) Find Z—i for the equation 2y + 2% = 2tan(y). (Xakf + ’Xl) = %:— (:LW(%\\ o ) 4 ) = e (o) 3%1\"5 *an\‘\q'éL + 2x = ZSecl(‘\\éiL o Ix (6«334 9 e (L‘)>d,;in 2"~ 1x ( Ay LAY dady-aly)
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MATH 1151 Test 11 - Page 4 of 5 November 15, 2024 3. Let f(z) =2+ v2z + L. (a) (2 points) Prove that f is a one-to-one function. (b) (3 points) Derive the formula for the inverse function f~(z). (¢) (2 points) Find the equation of the tangent line to the curve y = f “Yr)atz =4 ekt W = ). INAE V(- L>b NN e e froa cYavE oA meeng IXF = 2x2 ¥ 2%, = WK X 2R e )= Heg) dmglies 0= - (o) x=2+f2+| @67’(,(): 2x-2)
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MATH 1151 Test 11 - Page 5 of 5 November 15, 2024 4. (4 points) Compute the derivative of f(z) = z*. I [46) - e (X°) Im (§) = X Ua k) 4 (nle))> S (1lnb0) 1 < b+ X ,e(fl /?\()5 - ch(tn(x”l) 5. (4 points) The radius of a circular puddle is expanding at a rate of 1 cm/s. When the radius is 5 cm, find how fast the area of the puddle is increasing. Gron: dr = lonfs Pt dbe whan ToSeom o\ Joss A= gk = -2rde Ok ' = 7-2(5) -\ = 0w W75
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