# MTH 151 Study Guide - Midterm Guide: Brodmann Area 24, Differential Equation, If And Only If

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Published on 15 Sep 2018

Department

Mathematics

Course

MTH 151

Professor

Miami University

MTH 151

Calculus I

Winter 2018

Term Test 4

Exam Guide

Table of content

- Fermat’s Theorem

- Max & Min values

- Exponential Growth and Decay

- Rolle’s Theorem & the mean value theorem

- Integral test

- Optimization

- Indeterminant Forms

- Curve Sketching

- Graphing

- Antiderivatives

- Area Problem

- Fundamental theorem of calculus

- Indefinite integral

Fermat’s Theorem

Sketch of proof

Suppose C [a, b] is such that f(c) is a local maximum and f’(c) exists.

Since f’(c) exists, f’(c) = lim

()()

exists

If h near 0, c + h bear c. Since f(c) is a local maximum, f(c + h) f(c)

F(c + h) – f(c) 0

We want to examine the sign of lim

()()

Consider left & right hand limits

h 0- : f’(c) = lim

()()

= lim

()()

lim

0 = 0

as h 0- ()()

0 since h < 0 & f(c+h) – f(c) 0

therefore f’(c) 0

h 0+ , f(c) = lim

()()

= lim

()()

lim

0 = 0

f’(c) 0

Therefore since f’(c) 0 & f’(c) 0 , f’(c) = 0