MAT 295 Lecture 17: Finding the Derivatives of ArcSin, ArcCos and ArcTan

A useful remark: Since the derivative function is defined through limits, it is easy to show the following result: If f(s) and g(x) are equal on an open interval (a, b), and f?(c) exists for some c E (a, b), then g?(c) also exists and g(c) = f (c). this allows one to calculate the derivative of functions Defined piecewise, except at the x values where the definition changes from one formula to another. For example: if then h (x) = 4x on (- infinity , 5) (because it agrees with (2x^2 + 4)? there) and h?(x) = 2e^2x on (5, infinity) (because it agrees with (e^2x) there). Whether or not h?(5) exists requires more work and we will see how to answer this in the following two questions. (1) Consider the function (a) Show that f(s) is continuous at x = 0 by calculating the left and right hand limits of f(s) as it approaches 0 and showing they are both equal to f(0). (b) Show that f(s) is not differentiable at 0 by calculating and showing the two sided limit does not exist (note: calculate these limits explicitly, do not take shortcuts using derivative formulas). These one - sided limits are called the left and right - sided derivatives of f(x) at 0. By definition, f?(0) exists if and only if the two one - sided derivatives both exists and are equal (note: there is nothing special about 0; one sided derivatives are defined similarity for any s value). (c) Using the Power Rule Theorem from class and the remark preceding this question, calculate f(x) (note that by (b), 0 should not be in the domain of .f?(x)). Sketch a graph of f(s) and f(x) on the same XY - axis (you do not need to be precise, just sketch time shape).

TE:- 1 x on * : E Ecuation Smeets - x \CD Areas Formulee x \ D stetics TT 111617.x A Statics HW 23 C calculus cuestion X y WeaWork: 117|cp x 5 1/2 (1-4sin^3(CPU). Xo statics Product of x * - e o webworkiriath.ttu.edu/webwork2/11/jopadyetiri2450-002508/HW10 13.1-13.3/14/ Results for this submission Problems Entered Answer Preview Result incorrect Problem 1 / N correct Problem 27 Problem 3V incorrect Prublerri 47 incorrect Prublerri 5 Incorrect Prublerri 6 Problem 7 V At least one of the answers above is NOT correct. Problem 8 4 of the questions remain unanswered. Problem v Problem 10 (1 point) For each of the following vector fields F, decide whether it is conservative or not by computing curl F. Type in a potential function f (that is, Vf=F). If it is not conservative, type N. Problem 11/ Problem 12 A. F(x, y) = (10s – tiny) i+ (-6.x + 411)j Problem 13... f(x, y) = ['roblem 11... B. F(z,y) – 5yi izj Problem 15/ Problem 16V f(x, y) – N Problem 17. C.F(x, y, z) = 5xi + Byj+k Problem 18 f(x, y, z) = Problem 19. Problem 20 D. F(x, y) = (5 sin ey) i+ (-12y + 5.x.com y)j Problem 217 f(x, y) = Problem 22 E. F(t, y, z) = x'i-By’j + 2zk S(x, y, ) = Note: Your answers should be either expressions of X, y and Z (e.g. "3xy + 2yz"), or the letter "N" 10 Type here to search Note: You can earn Daniel Credit on this problem 0 E 9 * ^ I 4% 4- 1239 PM = 11242011

Ed 9th B Blackboarc x h, ed rds.pd Adobe Read Wind 178 1334 100% Tools Sign (b) If 0 is increasing at the rate of radian per minute, find the when rates of change of the area when 0 /6 and 6 m/3 Explain why the rate of change of the area of the triangle is when not constant even though d6/dt is constant. 16. Volume The radius of a sphere is increasing at a rate of when 3 inches per minute a) Find the rates of c hange of the volume when 9 inches when and 36 inches b) Explain why the rate of change of the volume of the sphere when is not constant even though dr/dt is constant. 17. Volume A spherical balloon is inflated with gas at the rate of when 3, y 800 cubic centimeters per minute. How fast is the radius of the balloon increasing at the instant the radius is a) 30 centi and (b) 60 centimeters when 4, y 18. Volume All edges of a cube are expanding at a ate of 6 centimeters per second. How fast is e volume c hanging In Exercises 5-8, a point is moving along the graph of the given hen each edge is (a) 2 centimeters and (b) 10 centimeters? function such that dxldt is 2 centimeters per second. Find dyldt for the given values of x 19. Surface Area The conditions are the same as Exercise 18 Determine how fast the surfa changing when each edge is (a) 2 centimeters and (b) 10 centimeter 20. Volume The formula for the volume of a cone is V Find the rates of change of the volume if dr/dt is 2 inches per minute and h 3r when (a 6 inches and (b 7. y 21. Volume At a sand and gravel plant sand is falling off a conveyor and onto a conical pile at a rate of 10 cubic feet per minute. The diameter of the base of the cone is approximately WRITING ABOUT CONCEPTS three times the altitude. At what rate is the heigh of the pile changing when the pile is 15 feet high? 9. Consider th e li near function y b. If changes at a constant rate, does y change at a constant rate? If so, does 22. Depth A conical tank (w th vertex down s 10 feet across th change at the same rate as x? Explain. top and 12 feet deep. If water is flowing into the tank at a rate of 10 cubic feet per minute, find the rate of change of the depth 10. In your ow words, state the guidelines for solving related of the water when th e water is 8 feet rate problems 23. Depth A swimming pool is 12 meters long, 6 meters wide 1 meter deep at the shallow end, and 3 meters deep at the deep 11. Find the rate of change of the distance between the origin end (see figure on next page Water is being pumped nto th f dx/di and a moving point n the graph of y pool at L cubic meter per minute. and there is 1 meter of water centimeters per second the deep end. 12. Find the rate of change of the distance between the origin (a) What percent of the pool is filled? f dx/dt and a moving point on the graph of y b) At what ate is th e water leve sing? centimeters per second 13. Area The radius of a circle is increasing at a rate of 4 centimeters per minute. Find the rates of change of the area. when (a 8 centimeters and (b 32 centimeters Comment a A Lueth 18 Research Help 6:30 PM 6/4/2013