# Calculus and Parametric Equations

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Calculus and Parametric Equations MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2018

2 Introduction Given a pair a parametric equations x = f (t) y = g(t) for a t b we know how to graph the parametric curve. y x

3 Objectives Today we will focus our attention on finding: the slope of the tangent line to the graph of a parametric curve, the area enclosed by a simple closed curve, the arc length of a parametric curve, and the surface area of a surface of revolution.

4 Slope of the Tangent Line Suppose x = f (t) and y = g(t), by the Chain Rule for Derivatives dy = dy dx dx. Let (x 0, y 0 ) = (x(t 0 ), y(t 0 )) then so long as dx (t 0) 0 then dy dx (x 0) = Remark: If x (t 0 ) = y (t 0 ) = 0 then dy (t 0) dx (t 0). dy dx (x 0) = lim t t0 provided the limit exists. dy dx = lim t t0 y (t) x (t),

5 Example Find the slope and equation of the tangent line for the following parametric equations at t = 1. x = t 3 t y = t 4 5t y x -1-2

6 Solution dx = 3t 2 1 dy = 4t 3 10t dy dx = 4 10 t=1 3 1 = 3 Since (x(1), y(1)) = (0, 0) then the equation of the tangent line is y = 3x

7 Find the Second Derivative (Concavity) The second derivative is the derivative of the first derivative. ( ) d 2 y dx 2 = d ( ) d dy dy dx = dx dx dx

8 Find the Second Derivative (Concavity) The second derivative is the derivative of the first derivative. ( ) d 2 y dx 2 = d ( ) d dy dy dx = dx dx Note: d 2 y dx 2 d 2 y 2 d 2 x 2 dx

9 Example Find d 2 y for the following parametric equations at t = 1. dx 2 x = t 3 t y = t 4 5t y x -1-2

10 Solution d 2 y dx 2 = = = d 2 y dx 2 = t=1 d d ( ) dy dx dx ( 4t 3 10t 3t 2 1 3t 2 1 ) (12t 2 10)(3t 2 1) (4t 3 10t)(6t) (3t 2 1) 2 3t 2 1 (12 10)(3 1) (4 10)(6) (3 1) 2 = 5 3 1

11 Finding Horizontal and Vertical Tangent Lines Theorem Suppose that x (t) and y (t) are continuous. Then for the curve defined by the parametric equations x = x(t) y = y(t) 1. if y (c) = 0 and x (c) 0, there is a horizontal tangent line at the point (x(c), y(c)). 2. if x (c) = 0 and y (c) 0, there is a vertical tangent line at the point (x(c), y(c)).

12 Example Find the points at which the graph of the following parametric equations has horizontal or vertical tangent lines. x = t 2 1 y = t 4 4t 2 30 y x -5

13 Solution dx dy = 2t = 4t 3 8t = 4t(t 2 2)

14 Solution dx dy = 2t = 4t 3 8t = 4t(t 2 2) Since when y (± 2) = 0 and x (± 2) = ±2 2 0 then the graph has horizontal tangents when t = ± 2. ( x(± 2), y(± ) 2) = (2 1, 4 8) = (1, 4)

15 Solution dx dy = 2t = 4t 3 8t = 4t(t 2 2) Since when y (± 2) = 0 and x (± 2) = ±2 2 0 then the graph has horizontal tangents when t = ± 2. ( x(± 2), y(± ) 2) = (2 1, 4 8) = (1, 4) Note that x (0) = y (0) = 0 so the slope of the tangent line when t = 0 is 4t 3 8t lim t 0 2t There are no vertical tangents. = lim t 0 2(t 2 2) = 4 0.

16 Velocity and Speed If the position of a moving object is given by the parametric equations x = x(t) y = y(t) where x(t) and y(t) are differentiable we say the horizontal component of velocity is given by x (t), the vertical component of velocity is given by y (t), and the speed is given by [x (t)] 2 + [y (t)] 2.

17 Example Find the components of velocity and the speed of an object moving according to the parametric equations at t = π/2. x = 3 cos t + sin 3t y = 3 sin t + cos 3t 4 y x -2-4

18 Solution x (t) = 3 sin t + 3 cos 3t y (t) = 3 cos t 3 sin 3t

19 Solution x (t) = 3 sin t + 3 cos 3t y (t) = 3 cos t 3 sin 3t x (π/2) = 3 y (π/2) = 3

20 Solution x (t) = 3 sin t + 3 cos 3t y (t) = 3 cos t 3 sin 3t x (π/2) = 3 y (π/2) = 3 s(π/2) = ( 3) 2 + (3) 2 = 3 2

21 Area Enclosed by a Curve (1 of 2) Recall: if y = f (x) 0 for a x b then the area under the curve, above the x-axis and between x = a and x = b is given by A = b a f (x) dx = b a y dx.

22 Area Enclosed by a Curve (1 of 2) Recall: if y = f (x) 0 for a x b then the area under the curve, above the x-axis and between x = a and x = b is given by A = b a f (x) dx = b a y dx. If the region is enclosed by parametrically defined curves with c t d then A = b a x = x(t) y = y(t) d y dx }{{}}{{} = y(t)x (t). y(t) x c (t)

23 Area Enclosed by a Curve (2 of 2) Theorem Suppose that the parametric equations x = x(t) and y = y(t) with c t d describe a curve that is traced out clockwise exactly once as t increases from c to d and where the curve does not intersect itself, except that the initial and terminal points are the same, i.e., x(c) = x(d) and y(c) = y(d). Then the enclosed area is given by A = d c d y(t)x (t) = x(t)y (t). c If the curve is traced out counterclockwise, then the enclosed curve is given by d d A = y(t)x (t) = x(t)y (t). c c

24 Example Find the area enclosed by the graph of the parametric curve described by for 0 t 2π. x = t sin t y = 1 cos t y x

25 Solution A = = = = 2π 0 2π 0 2π 0 2π 0 = 3π (1 cos t)(1 cos t) (1 2 cos t + cos 2 t) (1 2 cos t + 12 ) (1 + cos 2t) ( cos t + 1 ) cos 2t 2

26 Example The ellipse whose general formula is x 2 a 2 + y 2 = 1 for a, b > 0 b2 is described parametrically by x = a cos t y = b sin t for 0 t 2π. Use the parametric equations to find a formula for the area of an ellipse.

27 Solution A = 2π 0 2π = ab 0 = ab 2 = a b π 2π 0 (b sin t)( a sin t) sin 2 t (1 cos 2t)

28 Homework Read Section 10.2 Exercises: WebAssign/D2L

### Plane Curves and Parametric Equations

Plane Curves and Parametric Equations MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2018 Introduction We typically think of a graph as a curve in the xy-plane generated by the

### Arc Length and Surface Area in Parametric Equations

Arc Length and Surface Area in Parametric Equations MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2011 Background We have developed definite integral formulas for arc length

### 10.3 Parametric Equations. 1 Math 1432 Dr. Almus

Math 1432 DAY 39 Dr. Melahat Almus almus@math.uh.edu OFFICE HOURS (212 PGH) MW12-1:30pm, F:12-1pm. If you email me, please mention the course (1432) in the subject line. Check your CASA account for Quiz

### Speed and Velocity: Recall from Calc 1: If f (t) gives the position of an object at time t, then. velocity at time t = f (t) speed at time t = f (t)

Speed and Velocity: Recall from Calc 1: If f (t) gives the position of an object at time t, then velocity at time t = f (t) speed at time t = f (t) Math 36-Multi (Sklensky) In-Class Work January 8, 013

### D. Correct! This is the correct answer. It is found by dy/dx = (dy/dt)/(dx/dt).

Calculus II - Problem Solving Drill 4: Calculus for Parametric Equations Question No. of 0 Instructions: () Read the problem and answer choices carefully () Work the problems on paper as. Find dy/dx where

### AP Calculus (BC) Chapter 10 Test No Calculator Section. Name: Date: Period:

AP Calculus (BC) Chapter 10 Test No Calculator Section Name: Date: Period: Part I. Multiple-Choice Questions (5 points each; please circle the correct answer.) 1. The graph in the xy-plane represented

### Chain Rule. MATH 311, Calculus III. J. Robert Buchanan. Spring Department of Mathematics

3.33pt Chain Rule MATH 311, Calculus III J. Robert Buchanan Department of Mathematics Spring 2019 Single Variable Chain Rule Suppose y = g(x) and z = f (y) then dz dx = d (f (g(x))) dx = f (g(x))g (x)

### Parametric Curves. Calculus 2 Lia Vas

Calculus Lia Vas Parametric Curves In the past, we mostly worked with curves in the form y = f(x). However, this format does not encompass all the curves one encounters in applications. For example, consider

### Differentiation - Quick Review From Calculus

Differentiation - Quick Review From Calculus Philippe B. Laval KSU Current Semester Philippe B. Laval (KSU) Differentiation - Quick Review From Calculus Current Semester 1 / 13 Introduction In this section,

### Extrema of Functions of Several Variables

Extrema of Functions of Several Variables MATH 311, Calculus III J. Robert Buchanan Department of Mathematics Fall 2011 Background (1 of 3) In single-variable calculus there are three important results

### Figure: Aparametriccurveanditsorientation

Parametric Equations Not all curves are functions. To deal with curves that are not of the form y = f (x) orx = g(y), we use parametric equations. Define both x and y in terms of a parameter t: x = x(t)

### (1) Recap of Differential Calculus and Integral Calculus (2) Preview of Calculus in three dimensional space (3) Tools for Calculus 3

Math 127 Introduction and Review (1) Recap of Differential Calculus and Integral Calculus (2) Preview of Calculus in three dimensional space (3) Tools for Calculus 3 MATH 127 Introduction to Calculus III

### Stokes Theorem. MATH 311, Calculus III. J. Robert Buchanan. Summer Department of Mathematics. J. Robert Buchanan Stokes Theorem

tokes Theorem MATH 311, alculus III J. Robert Buchanan Department of Mathematics ummer 2011 Background (1 of 2) Recall: Green s Theorem, M(x, y) dx + N(x, y) dy = R ( N x M ) da y where is a piecewise

### MTH4100 Calculus I. Week 6 (Thomas Calculus Sections 3.5 to 4.2) Rainer Klages. School of Mathematical Sciences Queen Mary, University of London

MTH4100 Calculus I Week 6 (Thomas Calculus Sections 3.5 to 4.2) Rainer Klages School of Mathematical Sciences Queen Mary, University of London Autumn 2008 R. Klages (QMUL) MTH4100 Calculus 1 Week 6 1 /

### = cos(cos(tan t)) ( sin(tan t)) d (tan t) = cos(cos(tan t)) ( sin(tan t)) sec 2 t., we get. 4x 3/4 f (t) 4 [ ln(f (t)) ] 3/4 f (t)

Tuesday, January 2 Solutions A review of some important calculus topics 1. Chain Rule: (a) Let h(t) = sin ( cos(tan t) ). Find the derivative with respect to t. Solution. d (h(t)) = d (sin(cos(tan t)))

### Fundamental Theorem of Calculus

Fundamental Theorem of Calculus MATH 6 Calculus I J. Robert Buchanan Department of Mathematics Summer 208 Remarks The Fundamental Theorem of Calculus (FTC) will make the evaluation of definite integrals

### Mathematics Engineering Calculus III Fall 13 Test #1

Mathematics 2153-02 Engineering Calculus III Fall 13 Test #1 Instructor: Dr. Alexandra Shlapentokh (1) Which of the following statements is always true? (a) If x = f(t), y = g(t) and f (1) = 0, then dy/dx(1)

### Math 259 Winter Solutions to Homework # We will substitute for x and y in the linear equation and then solve for r. x + y = 9.

Math 59 Winter 9 Solutions to Homework Problems from Pages 5-5 (Section 9.) 18. We will substitute for x and y in the linear equation and then solve for r. x + y = 9 r cos(θ) + r sin(θ) = 9 r (cos(θ) +

### Conic Sections in Polar Coordinates

Conic Sections in Polar Coordinates MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2018 Introduction We have develop the familiar formulas for the parabola, ellipse, and hyperbola

### AP Calculus BC - Problem Solving Drill 19: Parametric Functions and Polar Functions

AP Calculus BC - Problem Solving Drill 19: Parametric Functions and Polar Functions Question No. 1 of 10 Instructions: (1) Read the problem and answer choices carefully () Work the problems on paper as

### Exam 3 Solutions. Multiple Choice Questions

MA 4 Exam 3 Solutions Fall 26 Exam 3 Solutions Multiple Choice Questions. The average value of the function f (x) = x + sin(x) on the interval [, 2π] is: A. 2π 2 2π B. π 2π 2 + 2π 4π 2 2π 4π 2 + 2π 2.

### 10.1 Curves Defined by Parametric Equation

10.1 Curves Defined by Parametric Equation 1. Imagine that a particle moves along the curve C shown below. It is impossible to describe C by an equation of the form y = f (x) because C fails the Vertical

### Green s Theorem. MATH 311, Calculus III. J. Robert Buchanan. Fall Department of Mathematics. J. Robert Buchanan Green s Theorem

Green s Theorem MATH 311, alculus III J. obert Buchanan Department of Mathematics Fall 2011 Main Idea Main idea: the line integral around a positively oriented, simple closed curve is related to a double

### Department of Mathematical and Statistical Sciences University of Alberta

MATH 4 (R) Winter 8 Intermediate Calculus I Solutions to Problem Set #5 Completion Date: Frida Februar 5, 8 Department of Mathematical and Statistical Sciences Universit of Alberta Question. [Sec.., #

### Math 106 Answers to Exam 3a Fall 2015

Math 6 Answers to Exam 3a Fall 5.. Consider the curve given parametrically by x(t) = cos(t), y(t) = (t 3 ) 3, for t from π to π. (a) (6 points) Find all the points (x, y) where the graph has either a vertical

### Math 113 Final Exam Practice

Math Final Exam Practice The Final Exam is comprehensive. You should refer to prior reviews when studying material in chapters 6, 7, 8, and.-9. This review will cover.0- and chapter 0. This sheet has three

### Parametric Functions and Vector Functions (BC Only)

Parametric Functions and Vector Functions (BC Only) Parametric Functions Parametric functions are another way of viewing functions. This time, the values of x and y are both dependent on another independent

### MATH 32A: MIDTERM 1 REVIEW. 1. Vectors. v v = 1 22

MATH 3A: MIDTERM 1 REVIEW JOE HUGHES 1. Let v = 3,, 3. a. Find e v. Solution: v = 9 + 4 + 9 =, so 1. Vectors e v = 1 v v = 1 3,, 3 b. Find the vectors parallel to v which lie on the sphere of radius two

### Chapter 10 Conics, Parametric Equations, and Polar Coordinates Conics and Calculus

Chapter 10 Conics, Parametric Equations, and Polar Coordinates 10.1 Conics and Calculus 1. Parabola A parabola is the set of all points x, y ( ) that are equidistant from a fixed line and a fixed point

### Taylor and Maclaurin Series

Taylor and Maclaurin Series MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2018 Background We have seen that some power series converge. When they do, we can think of them as

### Functions of Several Variables

Functions of Several Variables Partial Derivatives Philippe B Laval KSU March 21, 2012 Philippe B Laval (KSU) Functions of Several Variables March 21, 2012 1 / 19 Introduction In this section we extend

### 1 Antiderivatives graphically and numerically

Math B - Calculus by Hughes-Hallett, et al. Chapter 6 - Constructing antiderivatives Prepared by Jason Gaddis Antiderivatives graphically and numerically Definition.. The antiderivative of a function f

### Implicit Differentiation and Inverse Trigonometric Functions

Implicit Differentiation an Inverse Trigonometric Functions MATH 161 Calculus I J. Robert Buchanan Department of Mathematics Summer 2018 Explicit vs. Implicit Functions 0.5 1 y 0.0 y 2 0.5 3 4 1.0 0.5

### PARAMETRIC EQUATIONS AND POLAR COORDINATES

10 PARAMETRIC EQUATIONS AND POLAR COORDINATES PARAMETRIC EQUATIONS & POLAR COORDINATES We have seen how to represent curves by parametric equations. Now, we apply the methods of calculus to these parametric

### Math 190 (Calculus II) Final Review

Math 90 (Calculus II) Final Review. Sketch the region enclosed by the given curves and find the area of the region. a. y = 7 x, y = x + 4 b. y = cos ( πx ), y = x. Use the specified method to find the

### Separable Differential Equations

Separable Differential Equations MATH 6 Calculus I J. Robert Buchanan Department of Mathematics Fall 207 Background We have previously solved differential equations of the forms: y (t) = k y(t) (exponential

### Math 180, Final Exam, Spring 2008 Problem 1 Solution. 1. For each of the following limits, determine whether the limit exists and, if so, evaluate it.

Math 80, Final Eam, Spring 008 Problem Solution. For each of the following limits, determine whether the limit eists and, if so, evaluate it. + (a) lim 0 (b) lim ( ) 3 (c) lim Solution: (a) Upon substituting

### Dr. Allen Back. Sep. 10, 2014

Dr. Allen Back Sep. 10, 2014 The chain rule in multivariable calculus is in some ways very simple. But it can lead to extremely intricate sorts of relationships (try thermodynamics in physical chemistry...

### There are some trigonometric identities given on the last page.

MA 114 Calculus II Fall 2015 Exam 4 December 15, 2015 Name: Section: Last 4 digits of student ID #: No books or notes may be used. Turn off all your electronic devices and do not wear ear-plugs during

### HOMEWORK 3 MA1132: ADVANCED CALCULUS, HILARY 2017

HOMEWORK MA112: ADVANCED CALCULUS, HILARY 2017 (1) A particle moves along a curve in R with position function given by r(t) = (e t, t 2 + 1, t). Find the velocity v(t), the acceleration a(t), the speed

### SCORE. Exam 3. MA 114 Exam 3 Fall 2016

Exam 3 Name: Section and/or TA: Do not remove this answer page you will return the whole exam. You will be allowed two hours to complete this test. No books or notes may be used. You may use a graphing

### Exam 3. MA 114 Exam 3 Fall Multiple Choice Questions. 1. Find the average value of the function f (x) = 2 sin x sin 2x on 0 x π. C. 0 D. 4 E.

Exam 3 Multiple Choice Questions 1. Find the average value of the function f (x) = sin x sin x on x π. A. π 5 π C. E. 5. Find the volume of the solid S whose base is the disk bounded by the circle x +

### Chapter 3 Differentiation Rules

Chapter 3 Differentiation Rules Derivative constant function if c is any real number, then Example: The Power Rule: If n is a positive integer, then Example: Extended Power Rule: If r is any real number,

### Note: Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f (x) is a real number.

997 AP Calculus BC: Section I, Part A 5 Minutes No Calculator Note: Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers for which f () is a real number..

### Heat Equation on Unbounded Intervals

Heat Equation on Unbounded Intervals MATH 467 Partial Differential Equations J. Robert Buchanan Department of Mathematics Fall 28 Objectives In this lesson we will learn about: the fundamental solution

### Exercises for Multivariable Differential Calculus XM521

This document lists all the exercises for XM521. The Type I (True/False) exercises will be given, and should be answered, online immediately following each lecture. The Type III exercises are to be done

### Motion in Space. MATH 311, Calculus III. J. Robert Buchanan. Fall Department of Mathematics. J. Robert Buchanan Motion in Space

Motion in Space MATH 311, Calculus III J. Robert Buchanan Department of Mathematics Fall 2011 Background Suppose the position vector of a moving object is given by r(t) = f (t), g(t), h(t), Background

### 0, such that. all. for all. 0, there exists. Name: Continuity. Limits and. calculus. the definitionn. satisfying. limit. However, is the limit of its

L Marizzaa A Bailey Multivariable andd Vector Calculus Name: Limits and Continuity Limits and Continuity We have previously defined limit in for single variable functions, but how do we generalize this

### Final practice, Math 31A - Lec 1, Fall 2013 Name and student ID: Question Points Score Total: 90

Final practice, Math 31A - Lec 1, Fall 13 Name and student ID: Question Points Score 1 1 1 3 1 4 1 5 1 6 1 7 1 8 1 9 1 Total: 9 1. a) 4 points) Find all points x at which the function fx) x 4x + 3 + x

### MAT 122 Homework 7 Solutions

MAT 1 Homework 7 Solutions Section 3.3, Problem 4 For the function w = (t + 1) 100, we take the inside function to be z = t + 1 and the outside function to be z 100. The derivative of the inside function

### Absolute Convergence and the Ratio Test

Absolute Convergence and the Ratio Test MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2018 Bacground Remar: All previously covered tests for convergence/divergence apply only

### MATH CALCULUS I 2.2: Differentiability, Graphs, and Higher Derivatives

MATH 12002 - CALCULUS I 2.2: Differentiability, Graphs, and Higher Derivatives Professor Donald L. White Department of Mathematical Sciences Kent State University D.L. White (Kent State University) 1 /

MAC 232 Exam 3 Review Spring 209 This review, produced by the Broward Teaching Center, contains a collection of questions which are representative of the type you may encounter on the exam. Other resources

### Absolute Convergence and the Ratio Test

Absolute Convergence and the Ratio Test MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2018 Bacground Remar: All previously covered tests for convergence/divergence apply only

### Linear Independence. MATH 322, Linear Algebra I. J. Robert Buchanan. Spring Department of Mathematics

Linear Independence MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Introduction Given a set of vectors {v 1, v 2,..., v r } and another vector v span{v 1, v 2,...,

### MATH 2433 Homework 1

MATH 433 Homework 1 1. The sequence (a i ) is defined recursively by a 1 = 4 a i+1 = 3a i find a closed formula for a i in terms of i.. In class we showed that the Fibonacci sequence (a i ) defined by

### Practice Problems for Exam 3 (Solutions) 1. Let F(x, y) = xyi+(y 3x)j, and let C be the curve r(t) = ti+(3t t 2 )j for 0 t 2. Compute F dr.

1. Let F(x, y) xyi+(y 3x)j, and let be the curve r(t) ti+(3t t 2 )j for t 2. ompute F dr. Solution. F dr b a 2 2 F(r(t)) r (t) dt t(3t t 2 ), 3t t 2 3t 1, 3 2t dt t 3 dt 1 2 4 t4 4. 2. Evaluate the line

### Bessel s Equation. MATH 365 Ordinary Differential Equations. J. Robert Buchanan. Fall Department of Mathematics

Bessel s Equation MATH 365 Ordinary Differential Equations J. Robert Buchanan Department of Mathematics Fall 2018 Background Bessel s equation of order ν has the form where ν is a constant. x 2 y + xy

### NO CALCULATOR 1. Find the interval or intervals on which the function whose graph is shown is increasing:

AP Calculus AB PRACTICE MIDTERM EXAM Read each choice carefully and find the best answer. Your midterm exam will be made up of 5 of these questions. I reserve the right to change numbers and answers on

### Power Series. x n. Using the ratio test. n n + 1. x n+1 n 3. = lim x. lim n + 1. = 1 < x < 1. Then r = 1 and I = ( 1, 1) ( 1) n 1 x n.

.8 Power Series. n x n x n n Using the ratio test. lim x n+ n n + lim x n n + so r and I (, ). By the ratio test. n Then r and I (, ). n x < ( ) n x n < x < n lim x n+ n (n + ) x n lim xn n (n + ) x

### FINAL EXAM CALCULUS 2. Name PRACTICE EXAM

FINAL EXAM CALCULUS 2 MATH 2300 FALL 208 Name PRACTICE EXAM Please answer all of the questions, and show your work. You must explain your answers to get credit. You will be graded on the clarity of your

### SCORE. Exam 3. MA 114 Exam 3 Fall 2016

Exam 3 Name: Section and/or TA: Do not remove this answer page you will return the whole exam. You will be allowed two hours to complete this test. No books or notes may be used. You may use a graphing

### Volumes of Solids of Revolution Lecture #6 a

Volumes of Solids of Revolution Lecture #6 a Sphereoid Parabaloid Hyperboloid Whateveroid Volumes Calculating 3-D Space an Object Occupies Take a cross-sectional slice. Compute the area of the slice. Multiply

### MATH 162. Midterm 2 ANSWERS November 18, 2005

MATH 62 Midterm 2 ANSWERS November 8, 2005. (0 points) Does the following integral converge or diverge? To get full credit, you must justify your answer. 3x 2 x 3 + 4x 2 + 2x + 4 dx You may not be able

### Tangent and Normal Vectors

Tangent and Normal Vectors MATH 311, Calculus III J. Robert Buchanan Department of Mathematics Fall 2011 Navigation When an observer is traveling along with a moving point, for example the passengers in

### Math 212-Lecture 8. The chain rule with one independent variable

Math 212-Lecture 8 137: The multivariable chain rule The chain rule with one independent variable w = f(x, y) If the particle is moving along a curve x = x(t), y = y(t), then the values that the particle

### Vector Functions & Space Curves MATH 2110Q

Vector Functions & Space Curves Vector Functions & Space Curves Vector Functions Definition A vector function or vector-valued function is a function that takes real numbers as inputs and gives vectors

### NO CALCULATOR 1. Find the interval or intervals on which the function whose graph is shown is increasing:

AP Calculus AB PRACTICE MIDTERM EXAM Read each choice carefully and find the best answer. Your midterm exam will be made up of 8 of these questions. I reserve the right to change numbers and answers on

### Lecture for Week 6 (Secs ) Derivative Miscellany I

Lecture for Week 6 (Secs. 3.6 9) Derivative Miscellany I 1 Implicit differentiation We want to answer questions like this: 1. What is the derivative of tan 1 x? 2. What is dy dx if x 3 + y 3 + xy 2 + x

### 10.1 Review of Parametric Equations

10.1 Review of Parametric Equations Recall that often, instead of representing a curve using just x and y (called a Cartesian equation), it is more convenient to define x and y using parametric equations

### Math 131. The Derivative and the Tangent Line Problem Larson Section 2.1

Math 131. The Derivative and the Tangent Line Problem Larson Section.1 From precalculus, the secant line through the two points (c, f(c)) and (c +, f(c + )) is given by m sec = rise f(c + ) f(c) f(c +

### Tangent Lines and Derivatives

The Derivative and the Slope of a Graph Tangent Lines and Derivatives Recall that the slope of a line is sometimes referred to as a rate of change. In particular, we are referencing the rate at which the

### VANDERBILT UNIVERSITY. MATH 2300 MULTIVARIABLE CALCULUS Practice Test 1 Solutions

VANDERBILT UNIVERSITY MATH 2300 MULTIVARIABLE CALCULUS Practice Test 1 Solutions Directions. This practice test should be used as a study guide, illustrating the concepts that will be emphasized in the

### Mean Value Theorem. MATH 161 Calculus I. J. Robert Buchanan. Summer Department of Mathematics

Mean Value Theorem MATH 161 Calculus I J. Robert Buchanan Department of Mathematics Summer 2018 Background: Corollary to the Intermediate Value Theorem Corollary Suppose f is continuous on the closed interval

### Tuesday, Feb 12. These slides will cover the following. [cos(x)] = sin(x) 1 d. 2 higher-order derivatives. 3 tangent line problems

Tuesday, Feb 12 These slides will cover the following. 1 d dx [cos(x)] = sin(x) 2 higher-order derivatives 3 tangent line problems 4 basic differential equations Proof First we will go over the following

### Vector-Valued Functions

Vector-Valued Functions 1 Parametric curves 8 ' 1 6 1 4 8 1 6 4 1 ' 4 6 8 Figure 1: Which curve is a graph of a function? 1 4 6 8 1 8 1 6 4 1 ' 4 6 8 Figure : A graph of a function: = f() 8 ' 1 6 4 1 1

### Understanding Part 2 of The Fundamental Theorem of Calculus

Understanding Part of The Fundamental Theorem of Calculus Worksheet 8: The Graph of F () What is an Anti-Derivative? Give an eample that is algebraic: and an eample that is graphical: eample : Below is

### Mean Value Theorem. MATH 161 Calculus I. J. Robert Buchanan. Summer Department of Mathematics

Mean Value Theorem MATH 161 Calculus I J. Robert Buchanan Department of Mathematics Summer 2018 Background: Corollary to the Intermediate Value Theorem Corollary Suppose f is continuous on the closed interval

### Day 4: Motion Along a Curve Vectors

Day 4: Motion Along a Curve Vectors I give my stuents the following list of terms an formulas to know. Parametric Equations, Vectors, an Calculus Terms an Formulas to Know: If a smooth curve C is given

### Section 3.6 The chain rule 1 Lecture. Dr. Abdulla Eid. College of Science. MATHS 101: Calculus I

Section 3.6 The chain rule 1 Lecture College of Science MATHS 101: Calculus I (University of Bahrain) Logarithmic Differentiation 1 / 23 Motivation Goal: We want to derive rules to find the derivative

### Section 4.3 Concavity and Curve Sketching 1.5 Lectures. Dr. Abdulla Eid. College of Science. MATHS 101: Calculus I

Section 4.3 Concavity and Curve Sketching 1.5 Lectures College of Science MATHS 101: Calculus I (University of Bahrain) Concavity 1 / 29 Concavity Increasing Function has three cases (University of Bahrain)

### Section 14.1 Vector Functions and Space Curves

Section 14.1 Vector Functions and Space Curves Functions whose range does not consists of numbers A bulk of elementary mathematics involves the study of functions - rules that assign to a given input a

### 4.1 Analysis of functions I: Increase, decrease and concavity

4.1 Analysis of functions I: Increase, decrease and concavity Definition Let f be defined on an interval and let x 1 and x 2 denote points in that interval. a) f is said to be increasing on the interval

### Exam 1 Review SOLUTIONS

1. True or False (and give a short reason): Exam 1 Review SOLUTIONS (a) If the parametric curve x = f(t), y = g(t) satisfies g (1) = 0, then it has a horizontal tangent line when t = 1. FALSE: To make

### Parametric Curves You Should Know

Parametric Curves You Should Know Straight Lines Let a and c be constants which are not both zero. Then the parametric equations determining the straight line passing through (b d) with slope c/a (i.e.

### sec x dx = ln sec x + tan x csc x dx = ln csc x cot x

Name: Instructions: The exam will have eight problems. Make sure that your reasoning and your final answers are clear. Include labels and units when appropriate. No notes, books, or calculators are permitted

### AP Calculus BC 2013 Scoring Guidelines

AP Calculus BC Scoring Guidelines The College Board The College Board is a mission-driven not-for-profit organization that connects students to college success and opportunity. Founded in 9, the College

### Section 3.6 The chain rule 1 Lecture. Dr. Abdulla Eid. College of Science. MATHS 101: Calculus I

Section 3.6 The chain rule 1 Lecture College of Science MATHS 101: Calculus I (University of Bahrain) Logarithmic Differentiation 1 / 1 Motivation Goal: We want to derive rules to find the derivative of

### Homogeneous Equations with Constant Coefficients

Homogeneous Equations with Constant Coefficients MATH 365 Ordinary Differential Equations J. Robert Buchanan Department of Mathematics Spring 2018 General Second Order ODE Second order ODEs have the form

### Accelerating Convergence

Accelerating Convergence MATH 375 Numerical Analysis J. Robert Buchanan Department of Mathematics Fall 2013 Motivation We have seen that most fixed-point methods for root finding converge only linearly

### Math 116 Second Midterm March 20, 2013

Math 6 Second Mierm March, 3 Name: EXAM SOLUTIONS Instructor: Section:. Do not open this exam until you are told to do so.. This exam has 3 pages including this cover. There are 8 problems. Note that the

### Math 180 Written Homework Solutions Assignment #4 Due Tuesday, September 23rd at the beginning of your discussion class.

Math 180 Written Homework Solutions Assignment #4 Due Tuesday, September 23rd at the beginning of your discussion class. Directions. You are welcome to work on the following problems with other MATH 180

### Math 1310 Final Exam

Math 1310 Final Exam December 11, 2014 NAME: INSTRUCTOR: Write neatly and show all your work in the space provided below each question. You may use the back of the exam pages if you need additional space

### ENGI Parametric Vector Functions Page 5-01

ENGI 3425 5. Parametric Vector Functions Page 5-01 5. Parametric Vector Functions Contents: 5.1 Arc Length (Cartesian parametric and plane polar) 5.2 Surfaces of Revolution 5.3 Area under a Parametric

### MATH 423/ Note that the algebraic operations on the right hand side are vector subtraction and scalar multiplication.

MATH 423/673 1 Curves Definition: The velocity vector of a curve α : I R 3 at time t is the tangent vector to R 3 at α(t), defined by α (t) T α(t) R 3 α α(t + h) α(t) (t) := lim h 0 h Note that the algebraic

### Example 2.1. Draw the points with polar coordinates: (i) (3, π) (ii) (2, π/4) (iii) (6, 2π/4) We illustrate all on the following graph:

Section 10.3: Polar Coordinates The polar coordinate system is another way to coordinatize the Cartesian plane. It is particularly useful when examining regions which are circular. 1. Cartesian Coordinates

### MATH 100 Introduction to the Profession

MATH 100 Introduction to the Profession Differential Equations in MATLAB Greg Fasshauer Department of Applied Mathematics Illinois Institute of Technology Fall 2012 fasshauer@iit.edu MATH 100 ITP 1 What

### MATH Final Review

MATH 1592 - Final Review 1 Chapter 7 1.1 Main Topics 1. Integration techniques: Fitting integrands to basic rules on page 485. Integration by parts, Theorem 7.1 on page 488. Guidelines for trigonometric