The University of Memphis MATH 1920 Summer 2011 Calculus II Dwiggins FINAL EXAM There are 12 questions worth ten points each. # 1. Find the following antiderivatives:.

Slides:

Advertisements

Similar presentations

Unit 6 – Fundamentals of Calculus Section 6

Advertisements

Equation of a Tangent Line

Copyright © Cengage Learning. All rights reserved.

Homework Homework Assignment #11 Read Section 3.3 Page 139, Exercises: 1 – 73 (EOO), 71 Rogawski Calculus Copyright © 2008 W. H. Freeman and Company.

Derivative Review Part 1 3.3,3.5,3.6,3.8,3.9. Find the derivative of the function p. 181 #1.

7.1 Areas Between Curves To find the area: divide the area into n strips of equal width approximate the ith strip by a rectangle with base Δx and height.

Volume: The Disk Method

Chapter 6 – Applications of Integration

Rogawski Calculus Copyright © 2008 W. H. Freeman and Company As noted in Theorem 1, the sign of the second derivative on an interval indicates the concavity.

Integral calculus XII STANDARD MATHEMATICS. Evaluate: Adding (1) and (2) 2I = 3 I = 3/2.

Homework Homework Assignment #47 Read Section 7.1 Page 398, Exercises: 23 – 51(Odd) Rogawski Calculus Copyright © 2008 W. H. Freeman and Company.

Homework Homework Assignment #11 Read Section 6.2 Page 379, Exercises: 1 – 53(EOO), 56 Rogawski Calculus Copyright © 2008 W. H. Freeman and Company.

Warm Up Show all definite integrals!!!!! 1)Calculator Active: Let R be the region bounded by the graph of y = ln x and the line y = x – 2. Find the area.

Integration in polar coordinates involves finding not the area underneath a curve but, rather, the area of a sector bounded by a curve. Consider the region.

Full Year Review MTH 251/252/253 – Calculus. MTH251 – Differential Calculus Limits Continuity Derivatives Applications.

Calculus highlights for AP/final review

S C alculu. 1. Preliminaries 2. Functions and Limits 3. The Derivative 4. Applications of the Derivative 5. The Integral 6. Applications of the Integral.

10.2 – 10.3 Parametric Equations. There are times when we need to describe motion (or a curve) that is not a function. We can do this by writing equations.

Section 7.2a Area between curves.

6.3 Definite Integrals and the Fundamental Theorem.

Differential Equations: Slope Fields

THE FUNDAMENTAL THEOREM OF CALCULUS Section 4.4. When you are done with your homework, you should be able to… –Evaluate a definite integral using the.

7.4: The Fundamental Theorem of Calculus Objectives: To use the FTC to evaluate definite integrals To calculate total area under a curve using FTC and.

1 When you see… Find the zeros You think…. 2 To find the zeros...

Areas & Definite Integrals TS: Explicitly assessing information and drawing conclusions.

Conics, Parametric Equations, and Polar Coordinates

Chapter 10 – Parametric Equations & Polar Coordinates 10.2 Calculus with Parametric Curves 1Erickson.

The Fundamental Theorem of Calculus (4.4) February 4th, 2013.

4.4 The Fundamental Theorem of Calculus

5.4 Fundamental Theorem of Calculus Quick Review.

A PREVIEW OF CALCULUS Section 1.1. When you have finished your homework you should be able to… Understand what calculus is and how it compares with precalculus.

Sketching a Quadratic Graph Students will use equation to find the axis of symmetry, the coordinates of points at which the curve intersects the x-axis,

Solids of Revolution Disk Method

P roblem of the Day - Calculator Let f be the function given by f(x) = 3e 3x and let g be the function given by g(x) = 6x 3. At what value of x do the.

The Fundamental Theorem of Calculus

8.2 Area of a Surface of Revolution

Moments, Center of Mass, Centroids Lesson 7.6. Mass Definition: mass is a measure of a body's resistance to changes in motion  It is independent of a.

Derivatives Test Review Calculus. What is the limit equation used to calculate the derivative of a function?

What Does f’ Say About f? Many of the applications of calculus depend on our ability to deduce facts about a function f from information concerning its.

Final Review – Exam 4. Radius and Interval of Convergence (11.1 & 11.2) Given the power series, Refer to lecture notes or textbook of section 11.1 and.

Warm up Problems More With Integrals It can be helpful to guess and adjust Ex.

The Fundamental Theorem of Calculus is appropriately named because it establishes connection between the two branches of calculus: differential calculus.

1.1 Preview to calculus. Tangent line problem Goal: find slope of tangent line at P Can approximate using secant line First, let Second, find slope of.

4.3 Riemann Sums and Definite Integrals Definition of the Definite Integral If f is defined on the closed interval [a, b] and the limit of a Riemann sum.

Riemann sums & definite integrals (4.3) January 28th, 2015.

5.2 Riemann Sums and Area. I. Riemann Sums A.) Let f (x) be defined on [a, b]. Partition [a, b] by choosing These partition [a, b] into n parts of length.

Clicker Question 1 If x = e 2t + 1 and y = 2t 2 + t, then what is y as a function of x ? – A. y = (1/2)(ln 2 (x – 1) + ln(x – 1)) – B. y = ln 2 (x – 1)

A Preview of Calculus (1.1) September 18th, 2012.

5.3 Definite Integrals and Riemann Sums. I. Rules for Definite Integrals.

Multiple- Choice, Section I Part A Multiple- Choice, Section I Part B Free- Response, Section II Part A Free- Response, Section II Part B # of Questions.

C.2.5b – Volumes of Revolution – Method of Cylinders Calculus – Santowski 6/12/20161Calculus - Santowski.

Section 9.2: Parametric Equations – Slope, Arc Length, and Surface Area Slope and Tangent Lines: Theorem. 9.4 – If a smooth curve C is given by the equations.

CURVE SKETCHING The first application of derivatives we will study is using derivatives to determine the shape of the graph of a function. We will use.

Calculus 6-R Unit 6 Applications of Integration Review Problems.

4.1 NOTES. x-Axis – The horizontal line on the coordinate plane where y=0. y-Axis – The vertical line on the coordinate plane where x=0.

FRQ Review. Test Review Retakes by Wed FRQs 7(AB) or 10(BC) types 6 questions per year 9 points each Questions 1 and 2 – calculator Questions 3-6 – non.

Definite Integrals. Definite Integral is known as a definite integral. It is evaluated using the following formula Otherwise known as the Fundamental.

Chapter 6 The Definite Integral. There are two fundamental problems of calculus 1.Finding the slope of a curve at a point 2.Finding the area of a region.

The Fundamental Theorem of Calculus Area and The Definite Integral OBJECTIVES  Evaluate a definite integral.  Find the area under a curve over a given.

THE FUNDAMENTAL THEOREM OF CALCULUS Section 4.4. THE FUNDAMENTAL THEOREM OF CALCULUS Informally, the theorem states that differentiation and definite.

9.3: Calculus with Parametric Equations When a curve is defined parametrically, it is still necessary to find slopes of tangents, concavity, area, and.

Arc Length & Surfaces of Revolution (7.4)

Parametric equations Parametric equation: x and y expressed in terms of a parameter t, for example, A curve can be described by parametric equations x=x(t),

Homework, Page 139 Compute f ′ (x) using the limit definition.

Warm-Up! Find the average value of

The gradient of a tangent to a curve

2-4: Tangent Line Review &

Presentation transcript:

The University of Memphis MATH 1920 Summer 2011 Calculus II Dwiggins FINAL EXAM There are 12 questions worth ten points each. # 1. Find the following antiderivatives:. Part I – Techniques of Integration # 2. Calculate the following definite integrals: (a) (b) (a)

. Calculus II Page 2 Final Exam # 3. Find the following antiderivatives: (a) Part II – Geometric Calculations (b) # 4. (a) Sketch the region D bounded by the x-axis, the curve (b) Show that two integrals are required to calculate the area of D using vertical strips, while only one integral is required if horizontal strips are used to partition D. (c) Calculate the area of D using whichever method is easiest.

Page 3 # 6. Calculate the centroid of D, and use a theorem of Pappus to calculate the volume obtained when D is revolved about the line x = 4. Calculus IIFinal Exam # 5. Let D represent the region bounded by the y-axis, the line y = 4, and the curve x = y 2. Calculate the volume obtained when D is revolved about: (a) the x-axis (b) the y-axis

Page 4 Calculus IIFinal Exam (b) Let C denote the curve y = x 2, 0 < x < 1. Calculate the surface area obtained by revolving C about the y-axis. Part III – Series and Approximation # 8. Determine the interval of convergence for the power series

Page 5 Calculus IIFinal Exam # 9. Give the power series (with x 0 = 0) for each of the following functions, and give the interval of convergence for each series. (a) (b) (c) Part IV – Analytic Geometry # 10. Classify each of the following as the equation for an ellipse, a parabola, or a hyperbola. (a) (b) (c) (d) (e)

Page 6 Calculus IIFinal Exam # 11. Sketch the graph of the polar curve and calculate the area bounded by this curve. # 12. Consider the curve given parametrically by Fill in the following chart of values, indicating where the tangent line is either vertical or horizontal, and use this information to sketch the trajectory. txydy/dx –2 – What are the slopes of the tangent lines at the point where the curve crosses itself? x y Show that the second derivative tells us this curve is concave up for –1 < t < 1 and concave down otherwise.