Calculus and analytic geometry

Slides:

Advertisements

Similar presentations

Warm Up Determine the anti-derivative. Then differentiate your answer to check your work Evaluate the definite integral: 3.

Advertisements

Sec 3.1: Tangents and the Derivative at a Point

MA/CS 375 Fall MA/CS 375 Fall 2002 Lecture 29.

Rates of Change and Tangent Lines Section 2.4. Average Rates of Change The average rate of change of a quantity over a period of time is the amount of.

Integration – Adding Up the Values of a Function (4/15/09) Whereas the derivative deals with instantaneous rate of change of a function, the (definite)

Sec 3.2: The Derivative as a Function If ƒ’(a) exists, we say that ƒ is differentiable at a. (has a derivative at a) DEFINITION.

1 Instantaneous Rate of Change  What is Instantaneous Rate of Change?  We need to shift our thinking from “average rate of change” to “instantaneous.

3.3 –Differentiation Rules REVIEW: Use the Limit Definition to find the derivative of the given function.

Welcome to Physics C  Welcome to your first year as a “grown-up” (nearly!)  What are college physics classes like?  Homework for the AP student  Keeping.

2.1 Rates of Change and Limits A rock falls from a high cliff. The position of the rock is given by: After 2 seconds: average speed: What is the instantaneous.

2.1 Rates of Change and Limits Average and Instantaneous Speed –A moving body’s average speed during an interval of time is found by dividing the distance.

Rates of Change and Limits

+ Section Average velocity is just an algebra 1 slope between two points on the position function.

Index FAQ Introduction to Differentiation Tangents as Limits of Secant Lines Tangent Lines Linear Approximations of Functions Velocity Rates of Change.

A Solidify Understanding Task

Lesson 2-1: Rates of Change and Limits Day #1 AP Calculus Mrs. Mongold.

3.3 Rules for Differentiation. What you’ll learn about Positive Integer Powers, Multiples, Sums and Differences Products and Quotients Negative Integer.

Section 2.4b. The “Do Now” Find the slope of the given curve at x = a. Slope:

4.2 Mean Value Theorem Quick Review What you’ll learn about Mean Value Theorem Physical Interpretation Increasing and Decreasing Functions Other Consequences.

Velocity and Other Rates of Change Notes: DERIVATIVES.

Basic Differentiation Rules

Two kinds of rate of change Q: A car travels 110 miles in 2 hours. What’s its average rate of change (speed)? A: 110/2 = 55 mi/hr. That is, if we drive.

Lecture 12 Average Rate of Change The Derivative.

If f (x) is continuous over [ a, b ] and differentiable in (a,b), then at some point, c, between a and b : Mean Value Theorem for Derivatives.

Section 2.1 – Average and Instantaneous Velocity.

CHAPTER 1 : Introduction Limits and Continuity. What is Calculus? Mathematics of motion and change Whenever there is motion or growth, whenever there.

Lecture 5 Difference Quotients and Derivatives. f ‘ (a) = slope of tangent at (a, f(a)) Should be “best approximating line to the graph at the point (a,f(a))”

Finding the Derivative/Rate of Change.  The derivative of a constant is 0. That is, if c is a real number, then 1. Sketch a graph to demonstrate this.

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

2.2 Basic Differentiation Rules and Rate of Change

CALCULUS AND ANALYTIC GEOMETRY CS 001 LECTURE 03.

Table of Contents 1. Section 2.1 Rates of change and Limits.

PRODUCT & QUOTIENT RULES & HIGHER-ORDER DERIVATIVES (2.3)

2.3 Geometrical Application of Calculus

Product and Quotient Rules

MTH1170 Higher Order Derivatives

2.4 Rates of Change and Tangent Lines

Warm Up a) What is the average rate of change from x = -2 to x = 2? b) What is the average rate of change over the interval [1, 4]? c) Approximate y’(2).

Describing Motion with Equations

2.4 Rates of Change and Tangent Lines Day 1

BASIC DIFFERENTIATION RULES AND RATES OF CHANGE (2.2)

Rates of Change and Limits

EXTREMA and average rates of change

2.1 Rates of Change and Limits Day 1

Rates of Change and Limits

Rates of Change and Limits

Warm-Up: October 2, 2017 Find the slope of at.

Calculus I (MAT 145) Dr. Day Monday November 27, 2017

Sec 2.7: Derivative and Rates of Change

BASIC DIFFERENTIATION RULES AND RATES OF CHANGE (2.2)

AP Calculus Honors Ms. Olifer

Unit 6 – Fundamentals of Calculus Section 6

Chain Rule AP Calculus.

3.2: Differentiability.

2.2C Derivative as a Rate of Change

2 Differentiation 2.1 TANGENT LINES AND VELOCITY 2.2 THE DERIVATIVE

2.1 The Derivative and the Slope of a Graph

Rates of Change and Limits

Natural Logarithms - Differentiation

Sec 5.3: THE FUNDAMENTAL THEOREM OF CALCULUS

Packet #4 Definition of the Derivative

Derivatives: definition and derivatives of various functions

Math 180 Packet #20 Substitution.

5.2 Mean Value Theorem for Derivatives

Rates of Change and Tangent Lines

Packet #24 The Fundamental Theorem of Calculus

What is it? What good is it?

Rates of Change and Limits

Non-Uniform Motion Outcomes:

Calculus 3-7 CHAIN RULE.

Presentation transcript:

Calculus and analytic geometry Lecture 04 CS001 Calculus and analytic geometry

Average Rates of Change and Secant Lines

Rates of Change: Derivative at a Point

Example we studied the speed of a rock falling freely from rest near the surface of the earth. We knew that the rock fell y = 16t2 feet during the first t sec, and we used a sequence of average rates over increasingly short intervals to estimate the rock’s speed at the instant t = 1. What was the rock’s exact speed at this time?

Derivatives The process of calculating a derivative is called differentiation. To emphasize the idea that differentiation is an operation performed on a function y = ƒ(x), we use the notation

Differentiation Rules X ≠ 0

Differentiation Rules Example

Example Differentiate the following powers of x.

Example

Differentiation Rules

Example

Differentiation Rules

Differentiation Rules

Second- and Higher-Order Derivatives

Second- and Higher-Order Derivatives

Try this