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Calculus I

by: Darren Schulist

Calculus I MATH 1210

Darren Schulist
Utah State University
GPA 3.77


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This 12 page Class Notes was uploaded by Darren Schulist on Wednesday October 28, 2015. The Class Notes belongs to MATH 1210 at Utah State University taught by Staff in Fall. Since its upload, it has received 8 views. For similar materials see /class/230420/math-1210-utah-state-university in Mathematics (M) at Utah State University.

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Date Created: 10/28/15
MATH 1210 Calculus I l 27 Derivatives Our concept of instantaneous change is precisely one of the primary meanings of the derivative of a function Let s formally define the derivative The derivative of a function y fx at x a is fla i 01fahfa provided that the limit eXists Once again let P be a point on the curve of y f x at x a Then P has coordinates a fa Consider a nearby point determined by Qa h f a h We have y fx Qa hfa h Tangent Line Secant Line Pafa The slope of the line tangent to the curve at P is the derivative off at a Slope of Tangent Line at the point P m f39a zynolw MATH1210 Calculusl Examgle Find the equation ofthe line tangent to fx x2 r x at x 1 Sketch the graph ofy fx and the tangent line Solution We must rst find the slope ofthe line when x 1 f1hf1 f1LI lim 1 ham h 172hh217h72 hm Ha 7MM lmi h ummam mu h gm h 73 So the slope of the tangent line is m 3 The equation ofthe tangent line has the form y r3 b Substituting in the coordinates of the point 1f1 12 we get 27371b 23b 571 So the line is given by y r3 7 1 See the graph at right 2 mm 1 quot ofthetangentquot L r of fxhxatx1 MATH 1210 Calculus I The derivative has the following interpretations o The Limit of Slopes of Secant Lines 0 Slope of a Tangent Line 0 Instantaneous Rate of Change Examples of Instantaneous Rate of Change Velocity change in position Acceleration change in velocity Jerk change in acceleration Growth rates population stocks in ation taxes cancer cells etc o Rainfall snowfall wind velocity change in volume change in pressure etc Observations o The derivative instantaneous rate of change of a line linear function is the m o The derivative instantaneous rate of change of a constant function horizontal line is w o Wherever a continuous function is increasing the derivative is positive 0 Wherever a continuous function is increasing the derivative is negative Notation The notation used to refer to the derivative of a function varies For the moment we will represent the derivative using prime notation f 39a MATH 1210 Calculus I 26 Tangents Velocities and Other Rates of Change We begin to formalize our concept of instantaneous change by defining the slope of a tangent line to a curve at a point Let P be a point on the curve of y f x at x a Then P has coordinates a fa Let h be a small value near zero We consider the point near P determined by Qa h f a h y fx Qa hfa h Pafa What is the slope of the line segment secant line connecting points P and Q fah fa fah fa ah a h Slope We are ready to define the slope of the line tangent to the curve at P Not surprisingly this will require a limit Slope of Tangent Line at the point P m lim haO fahfa h MATH 1210 Calculus I Example Suppose a ball is dropped from the top of a radio tower 450 feet above the ground The distance traveled by the ball after I seconds is given by st 16t2 What is the velocity of the ball t 2 seconds after being dropped Solution Letting a 2 we have Velocity Slope of Tangent Line 2 MW haO 1 162 h2 1622 2 lim haO h 2 lim1644h h 64 haO h 64 64h 16h2 64 lim hgt0 h limh64 16h haO h i111364 16h 64 feet per second MATH 1210 Calculus I 0 Personal Intro Day 1 Calculus I Fall 2006 Fall 2004 0 PhD in Mathematics from the University of Wyoming 0 Currently work with Math Education Program 0 5 years in industry working on the Early Warning System DSP o Interests cycling desert climbing and big walls juggling photography 0 Course Intro 0 Syllabus Calculus 7 the language of motion and change Overview Interactive studentcentered class Expectations come to class work hard learn have fun Homework and Quizzes Read your text prior to coming to class each day gt Read it again after class for reinforcement it hopefully will make more sense gt Complete and re ect on homework Exams Perfect Exams No Homework 71 C Study Groups are effective Office hours 7 use them 7 these are also effective Review and know your trigonometry insert page 2 inside of front cover or Appendix C page A18 In light of struggles with trig by students in the past review this material NOW Specifically know radian measure and the table of function values in the middle of page A21 by Friday We will use nearly every trig relationship and fact from these pages by the end of the semester Therefore we will have daily quizzes on the fundamentals daily for the next two weeks or until the class gets a perfect score MATH 1210 Calculus I Fall 2006 Chapter 2 Limits and Derivatives For those who have had calculus in high school we will develop nearly every result from scratch While this is not an advanced calculus course we do want you to develop a deeper understanding of the concepts than most high school courses provide You will be expected to memorize some de nitions and work basic problems from these de nitions 21 Tangent and Velocitv Problems 0 If Jamie rides 123 miles in 6 hours and 13 minutes estimate her average speed to the nearest tenth What operational relationship is implied by the units 0 Beaver Mountain has a vertical drop of 1600 feet elevation over a trail that is 2 miles long Estimate the average slope of the trail What are the units Why Integpreting Graph Exercises a If the following graphs represent position over time describe the motion being represented i Sketch a graph of the corresponding velocity function ii Sketch the graph of the corresponding acceleration function b If the following graphs represent velocity over time describe the position of the particle i Sketch a graph of the position of the particle over time What is the initial position of the particle ii Sketch the graph of the corresponding acceleration function Graph 1 Graph 2 MATH 1210 Calculus I Fall 2006 21 Tangent and Velocity Problems cont Be sure to READ each section prior to coming to class Re reading each section is strongly suggested to reinforce concept and the notation Example Heartbeats per minute Given the following table tmin 36 38 40 42 44 IHeartbeatsI 2530 2661 2806 2948 3080 Estimate the patient s heart rate between the given values of t Note that Number of Heartbeats Length of Time Interval a t 36 and t 42 HR w 697 beats per minute 42 36 bt38 andt42 HRM7L75 bpm 42 38 ct40andt42 HRM71bpm 42 40 dt42andt44 HRM66bpm 44 42 Is there much uctuation in the patient s heart rate What would you estimate the patient s heart rate to be after at 42 minutes Average Rate of Change The average rate of change in a function f x is the slope of a line segment whose endpoints are x f x and x h f x h That is fx 10 fx fx 10 fx h xh x The instantaneous rate of change in a function f x is a measure of steepness 0f the graph of y fx at a specific point QUESTION How can we measure the slope ofa curve MATH 1210 Ca1eu1us1 Fall 2006 Example Posmon of Car 1 seconds 0 1 1 2 3 1 4 5 s feet I 0 1 10 32 70 1 119 178 a Fmd the average velomty for the ume penod begrhhrhg when t 2 and 1ashhg 787 h 2 seconds hr 1 second rr a a 2 Example Let x 27 x2 and consxderthe slope of the graph of y x at the pomt 12 11 Usrhs a secant hhe through the pomts P11 and Q2r2 we have 1 P11 111s apparent that the secant hhe 15 steeperthan the graph anhe pomt P Choosmg a pomt Q closer to poth we have thtifwe chase Q ehmsr m pniml MATH 1210 Calculus I Fall 2006 A Generalized Illustration Example Let fx x2 2xl Recall that f 3 32 2 31 fw w2 2wl fxhxh2 2xh1 Now express and simplify W What does this represent and what does it mean for h gt 0 To Summarize Section 21 0 Average rate of change is the slope of a secant line 0 The steepness of a graph at a point is the instantaneous rate of change 0 Instantaneous rate of change at a point P is the slope of the line tangent to the graph at the point P o In an applied context the units of rate of change beats per minute miles per hour etc reveal the proper ratio of the rate of change MATH 1210 Calculus I Fall 2006 22 The Limit of a Function Introductory discussion and examples 0 Regular Polygons limit of ngons as n gets large 0 9 99 999 9999 What is the value of 0999 o 1 15 175 1875 19375 o 3 31 314 3141 31415 314159 3141592 x99 2 o What is lim Hz x3 Questions 0 What do you expect the limit to be for a function whose graph is defined at x a and unbroken o What if the graph of a function is broken at x a o What if a function is unde ned at x a MATH 1210 Calculus I Fall 2006 MEMORIZE the definition of limit We write lim f x L and we say the limit of fx as x approaches a equals L if we can make the values of fx arbitrarily close to L as close to L as we like by taking x sufficiently close to a on either side of a but not equal to a We write lim f x L and we y the limit of fx as x approaches a equals L if we can make the values of fx arbitrarily close to L by taking x sufficiently close to a but not equal to a


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