Published on 28 Feb 2018

School

Department

Course

Professor

University of Toronto

Scarborough

MATA32H3

Calculus for Management I

Fall 2017

Final Exam

Exam Guide

CONTENTS

Exam Guide Covers the Following Topics:

•Derivatives and Concavity

•Absolute Extrema on a Closed and Bounded Interval

•Horizontal &Vertical Asymptotes

•Applied Optimization Problems

• Integration

•Integration by Parts

•The Definite Integral

•The Fundamental Theorem of Calculus

•Properties of the Definite Integral

•Applications of Integral to Area

•Areas Bounded Between Curves

1st Derivative Test

Idea: given y=f(x), the 1st derivative is used to determine whether a critical number of a

function is a local max, min, or neither

*** Critical number is 0 or undefined

Let y=f(x) be a function and assume ‘c’ is a critical number of ‘y’. Assume (I) is an open

interval

1. Max: if f’(x) changes signs from + to -, as ‘x’ moves through I, the ‘f’ has a local

max at ‘c’

2. Min: If f’(x) changes sign from + to -, as ‘x’ moves through I, the ‘f’ has a local

min at ‘c’

3. Neither: if f’(x) does not change sign for ‘x’ to the left side of ‘c’ in I, then ‘f’ has

no local max/min at ‘c’

2nd Derivative

Gives information about the bending upwards/downwards of a function

(i.e. concavity (point of inflection)

Concavity Test

Suppose we have a function y=f(x) so the f’’(x) is defined on a given open interval (a,b)

!I

1. if f’’(x) > 0 for all x of I, the f(x) is concave up on I

2. If f’’(x) <0 for all x of I, then f(x) is concave down on I

We obtain intervals I, as in the concavity test

1. Find f’’(x)

2. Solve f’(x)=0

3. See if f’’(x) is ever undefined

Derivatives and Concavity