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The Idea of Limits x1.91.991.9991.999922.00012.0012.012.1 f(x)f(x)
The Idea of Limits x1.91.991.9991.999922.00012.0012.012.1 f(x)f(x)3.93.993.9993.9999un- defined 4.00014.0014.014.1
The Idea of Limits x1.91.991.9991.999922.00012.0012.012.1 g(x)g(x)3.93.993.9993.999944.00014.0014.014.1 x y O 2
approaches to, but not equal to
The Idea of Limits x-4-3-201234 g(x)g(x)
The Idea of Limits x-4-3-201234 h(x)h(x) un- defined 1234
does not exist.
Limits at Infinity Consider
Generalized, if then
Theorems of Limits at Infinity
Contoh - contoh
The Slope of the Tangent to a Curve
The slope of the tangent to a curve y = f(x) with respect to x is defined as provided that the limit exists.
The increment △ x of a variable is the change in x from a fixed value x = x 0 to another value x = x 1.
(A) Definition of Derivative. The derivative of a function y = f(x) with respect to x is defined as provided that the limit exists.
Contoh - contoh
Limits and Derivatives Concept of a Function y is a function of x, and the relation y = x 2 describes a function. We notice that with such a relation,
Derivatives - Equation of the Tangent Line Now that we can find the slope of the tangent line of a function at a given point, we need to find the equation.
3.1 –Tangents and the Derivative at a Point The limiting value of the ratio of the change in a function to the corresponding change in its independent.
THE DERIVATIVE AND THE TANGENT LINE PROBLEM Section 2.1.
Sec 3.1: Tangents and the Derivative at a Point The difference quotient of ƒ at x 0 with increment h. Example: Find the difference quotient of ƒ at x0=2.
Business Calculus Derivative Definition. 1.4 The Derivative The mathematical name of the formula is the derivative of f with respect to x. This is the.
GOAL: USE DEFINITION OF DERIVATIVE TO FIND SLOPE, RATE OF CHANGE, INSTANTANEOUS VELOCITY AT A POINT. 3.1 Definition of Derivative.
The Derivative and the Tangent Line Problem. Local Linearity.
1.6 – Tangent Lines and Slopes Slope of Secant Line Slope of Tangent Line Equation of Tangent Line Equation of Normal Line Slope of Tangent =
Unit 2 Lesson #1 Derivatives 1 Interpretations of the Derivative 1. As the slope of a tangent line to a curve. 2. As a rate of change. The (instantaneous)
Implicit Differentiation. Objectives Students will be able to Calculate derivative of function defined implicitly. Determine the slope of the tangent.
Suppose that functions f and g and their derivatives have the following values at x = 2 and x = –4 1/3–3 5 Evaluate the derivatives with.
CHAPTER Continuity Derivatives Definition The derivative of a function f at a number a denoted by f’(a), is f’(a) = lim h 0 ( f(a + h) – f(a))
3.1 Definition of the Derivative & Graphing the Derivative.
Derivatives Using the Limit Definition At the end of this lesson you should be able to –use the limit definition to find the derivative of a function.
Objectives Use the first-order derivative to find the stationary points of a function. Use the second-order derivative to classify the stationary points.
Chapter 3.1 Tangents and the Derivative at a Point.
2.1 The Derivative and The Tangent Line Problem Slope of a Tangent Line.
Equations of Tangent Lines April 21 st & 22nd. Tangents to Curves.
The Derivative Chapter 3:. What is a derivative? A mathematical tool for studying the rate at which one quantity changes relative to another.
Definition of the Derivative Using Average Rate () a a+h f(a) Slope of the line = h f(a+h) Average Rate of Change = f(a+h) – f(a) h f(a+h) – f(a) h.
Section 15.3 Partial Derivatives. PARTIAL DERIVATIVES If f is a function of two variables, its partial derivatives are the functions f x and f y defined.
2.1 The Derivative and The Tangent Line Problem The Definition of a Derivative.
DO NOW: Find the equation of the line tangent to the curve f(x) = 3x 2 + 4x at x = -2.
Derivative Review Part 1 3.3,3.5,3.6,3.8,3.9. Find the derivative of the function p. 181 #1.
The derivative of a function f at a fixed number a is In this lesson we let the number a vary. If we replace a in the equation by a variable x, we get.
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.
Aim: How do we find the derivative by limit process? Do Now: Find the slope of the secant line in terms of x and h. y x (x, f(x)) (x + h, f(x + h)) h.
Tangent lines Recall: tangent line is the limit of secant line The tangent line to the curve y=f(x) at the point P(a,f(a)) is the line through P with slope.
Sec 2.7: DERIVATIVES AND RATES OF CHANGE Example: Find the derivative of the function at x = 2. Find Example: Find the derivative of the function at a.
DIFFERENTIATION & INTEGRATION CHAPTER 4. Differentiation is the process of finding the derivative of a function. Derivative of INTRODUCTION TO DIFFERENTIATION.
Warm Up. Equations of Tangent Lines September 10 th, 2015.
Every slope is a derivative. Velocity = slope of the tangent line to a position vs. time graph Acceleration = slope of the velocity vs. time graph How.
2.1 The Derivative and the Tangent Line Problem.
Chapter 3.2 The Derivative as a Function. If f ’ exists at a particular x then f is differentiable (has a derivative) at x Differentiation is the process.
§3.2 – The Derivative Function October 2, 2015.
Tangent Line Problems Find the equations of tangents at given points Find the points on the curve if tangent slope is known Find equations of tangents.
Application of Derivative - 1 Meeting 7. Tangent Line We say that a line is tangent to a curve when the line touches or intersects the curve at exactly.
2.5 Implicit Differentiation. Implicit and Explicit Functions Explicit FunctionImplicit Function But what if you have a function like this…. To differentiate:
Equations of Tangent Lines. Objective To use the derivative to find an equation of a tangent line to a graph at a point.
In Section 2.1 we considered the derivative at a fixed number a. Now let number a vary. If we replace number a by a variable x, then the derivative can.
Unit 6 – Fundamentals of Calculus Section 6.4 – The Slope of a Curve No Calculator.
The Derivative Calculus. At last. (c. 5). POD Review each other’s answers for c. 4: 23, 25, and 27.
Partial Derivatives. 1. Find both first partial derivatives (Similar to p.914 #9-40)
The derivative of f at x is given by f’(x) = lim f(x + ∆x) – f(x) ∆x -> 0 ∆x provided the limit exists. For all x for which this limit exists, f’ is a.
Calculus Section 3.1 Calculate the derivative of a function using the limit definition Recall: The slope of a line is given by the formula m = y 2 – y.
Homework questions? 2-5: Implicit Differentiation ©2002 Roy L. Gover (www.mrgover.com) Objectives: Define implicit and explicit functions Learn.
Example 4 Equation of a Tangent Line Chapter 8.5 Calculus can be used to find the slope of the tangent to a curve, and then we can write the equation of.
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