Derivative Methods: (csc(x)) = csc(x) cot(x)

Transcription

1 EXAM 2 IS TUESDAY IN QUIZ SECTION Allowe:. A Ti-30x IIS Calculator 2. An 8.5 by inch sheet of hanwritten notes (front/back) 3. A pencil or black/blue pen Covers: , 0.2, 3.9, 3.0, 4. Quick Review This test is about two things: A) All Derivative Methos B) Immeiate concepts an applications of erivatives See my newsletters (an homework) for targete practice problems on any of topic. I will take some problems irectly from homework. Applications an Concepts:. Geometric Tangent Slope/Line Questions 2. Linear approximation 3. Relate Rates!! 4. Fining Critical Numbers 5. Fining Absolute Max/Min Derivative Methos: 6. Power Rule: 7. Exponential Rules: x (xn ) = nx n 8. Prouct, Quotient, Chain Rules!! 9. Trig erivatives: (sin(x)) = cos (x) x x (tan(x)) = sec2 (x) (sec(x)) = sec(x) tan(x) x 0. Parametric Equation Derivatives y x = y/t = slope of tangent x/t ( x t ) 2 + ( y t ) 2 = spee. Implicit Differentiation!! x (ex ) = e x, x (ax ) = a x ln(a) (cos(x)) = sin (x) x x (cot(x)) = csc2 (x) (csc(x)) = csc(x) cot(x) x 2. Inverse Trig erivatives x (sin (x)) = x 2 x (cos (x)) = x 2 x (tan (x)) = + x 2 x (cot (x)) = + x 2 x (sec (x)) = x x 2 x (csc (x)) = x x 2 3. Deriv. of Logarithms: x (ln(x)) = x 4. Logarithmic Differentiation, x (log a(x)) = x ln(a)

2 Name: Math 24 - Winter 207 Exam 2 February 2, 207 Section: Stuent ID Number: PAGE 4 PAGE 2 2 PAGE 3 4 PAGE 4 8 PAGE 5 2 Total 60 There are 5 pages of questions. Make sure your exam contains all these questions. You are allowe to use a Ti-30x IIS Calculator moel ONLY (no other calculators allowe). An you are allowe one han-written 8.5 by inch page of notes (front an back). Leave your answer in exact form. Simplify stanar trig, inverse trig, natural logarithm, an root values. Here are several examples: you shoul write 4 = 2 an cos ( ) π 6 = 3 an 7 3 = an ln() = 0 an tan () = arctan() = π. 4 Show your work on all problems. The correct answer with no supporting work may result in no creit. Put a box aroun your FINAL ANSWER for each problem an cross out any work that you on t want to be grae. If you nee more room, use backs of the pages an inicate to the graer that you have one so. Raise your han if you have a question. There may be multiple versions of the exam so if you copy off a neighbor an put own the answers from another version we will know you cheate. Any stuent foun engaging in acaemic misconuct will receive a score of 0 on this exam. All suspicious behavior will be reporte to the stuent misconuct boar. DO NOT CHEAT OR DO ANYTHING THAT LOOKS SUSPICIOUS! WE WILL REPORT YOU AND YOU MAY BE EXPELLED! Keep your eyes own an on your paper. If your TA sees your eyes wanering they will warn you only once before taking your exam from you. You have 80 minutes to complete the exam. Buget your time wisely. SPEND NO MORE THAN 0 MINUTES PER PAGE! GOOD LUCK!

3 . (4 pts) (You on t nee to simplify your erivatives) (a) Let y = tan 5 (e 3x ). Fin y x. (b) Let g(x) = x 2 + arctan(2x). Fin the value(s) of x at which the slope of the tangent line to g(x) is. (c) Fin the value(s) of x at which f(x) = 4x x 4/3 has a horizontal tangent line.

4 2. (2 pts) For all parts below, consier a particle moving in the xy-plane such that its location at time t secons is given by: where x an y are in feet. (a) Fin the following: x(t) = 3 t3 7 2 t2 + 0t + 2, y(t) = ln(3t + ) + 4 t ln(4)t + 5, i. The formula for the vertical velocity in terms of time t. ii. The spee of the particle at time t = 0. (inclue units) iii. The equation for the tangent line to the curve at time t = 0 (in the form y = mx + b). (b) Fin all times, t, then the curve has a vertical tangent line.

5 3. (4 pts) (a) Fin the equation of the tangent line to y = (2x + ) cos(πx) at x =. (b) The implicit efine curve (x 2 +y) 2 +xy 5 = 4 has only one point where it crosses the positive y-axis. Fin the equation of the tangent line at this positive y-intercept.

6 4. (8 pts) A spherical snowball is melting. At the moment when the raius is 5 cm, its surface area is ecreasing at a rate of 3 cm 2 /min. Fin the rate at which the volume is changing at this same moment. (Inclue units in your final answer an inicate if your answer is positive or negative). Recall: Volume of a sphere is V = 4 3 πr3 an the surface area of a sphere is S = 4πr 2.

7 5. (2 pts) A kite is in the air at an altitue of 400 feet an is being blown horizontally at the constant rate of 0 feet per secon away from the person holing the kite string at groun level. (Thus, the kite is remaining at a constant altitue of 400 feet). For both parts: Inclue units for your final answers an inicate if your answers are positive or negative. Your final answers shoul be simplifie numbers/fractions. (a) At what rate is the string being let out when 500 feet of string is alreay out? (b) Let θ be the angle the string makes with the groun at a given time. At what rate is θ changing at the instant when 500 feet of string is alreay out?