Lesson3.1 The Derivative And The Tangent Line
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This document provides an introduction to the concept of the tangent line and derivative. It defines the tangent line as the line that intersects a curve at exactly one point. It discusses how to approximate the slope of a tangent line using secant lines and taking the limit as the second point approaches the point of tangency. The derivative is defined as the formula that gives the slope of the tangent line at any point on a curve. It provides examples of using calculators to calculate derivatives and discusses how the graph of a derivative relates to properties of the original function such as maxima, minima and x-intercepts.
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This document introduces the concept of the derivative and rate of change in mathematics. It defines the derivative as the slope of the tangent line to a curve at a point, which can be interpreted as the instantaneous rate of change of the dependent variable with respect to the independent variable. The document provides examples of calculating derivatives and interpreting them in the context of rates of change. It then discusses how to view the derivative not just at a single point, but as a new function defined for all points where the limit exists, known as the derivative function.Tam 2nd
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The document discusses techniques for finding the slope of a tangent line to a function at a given point using derivatives. It provides examples of finding the slope for various functions, including polynomials, trigonometric functions, and implicitly defined functions, using the definition of derivative and rules like the product rule. Approximations of slope using finite differences are also covered. Guidelines for performing implicit differentiation are outlined.3.1 Ppt
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This document discusses notation and definitions related to derivatives. It notes there are multiple ways to write derivatives, such as "f prime x" or "the derivative of f with respect to x". It also clarifies that dx and dy do not represent multiplication. A derivative represents the slope of a function and must be defined at endpoints for functions on closed intervals. For a function to have a derivative everywhere it must be continuous and smooth.Differentiation using First Principle - By Mohd Noor Abdul Hamid
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1. The document discusses the concept of the derivative and differentiation using the first principle. It explains how to calculate the slope of a tangent line to a curve at a point using limits, which gives the derivative of the function at that point.
2. Rules for differentiating common functions like polynomials, exponentials, and logarithms are covered. Higher-order derivatives and applications of derivatives to business and economics are also mentioned.
3. The goals of the class are to explain the concept of the derivative, differentiate functions using the first principle (limits), and understand various differentiation rules.Ad
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Lesson3.1 The Derivative And The Tangent Line
- 1. The Derivative and the Tangent Line Problem Lesson 3.1
- 3. Tangent Definition From geometry a line in the plane of a circle intersects in exactly one point We wish to enlarge on the idea to include tangency to any function, f(x)
- 4. Slope of Line Tangent to a Curve Approximated by secants two points of intersection Let second point get closer and closer to desired point of tangency View spreadsheet simulation
- 6. Slope of Line Tangent to a Curve Recall the concept of a limit from previous chapter Use the limit in this context
- 7. Definition of a Tangent Let x shrink from the left
- 8. Definition of a Tangent Let x shrink from the right
- 9. The Slope Is a Limit Consider f(x) = x 3 Find the tangent at x 0 = 2 Now finish
- 11. Calculator Capabilities Able to draw tangent line Steps Specify function on Y= screen F5-math, A-tangent Specify an x (where to place tangent line) Note results
- 12. Difference Function Creating a difference function on your calculator store the desired function in f(x) x^3 -> f(x) Then specify the difference function (f(x + dx) f(x))/dx -> difq(x,dx) Call the function difq(2, .001) Use some small value for dx Result is close to actual slope
- 13. Definition of Derivative The derivative is the formula which gives the slope of the tangent line at any point x for f(x) Note: the limit must exist no hole no jump no pole no sharp corner A derivative is a limit !
- 14. Finding the Derivative We will (for now) manipulate the difference quotient algebraically View end result of the limit Note possible use of calculator limit ((f(x + dx) f(x)) /dx, dx, 0)
- 15. Related Line the Normal The line perpendicular to the function at a point called the normal Find the slope of the function Normal will have slope of negative reciprocal to tangent Use y = m(x h) + k
- 16. Using the Derivative Consider that you are given the graph of the derivative What might the slope of the original function look like? Consider what do x-intercepts show? what do max and mins show? f `(x) <0 or f `(x) > 0 means what? To actually find f(x), we need a specific point it contains f `(x)
- 17. Derivative Notation For the function y = f(x) Derivative may be expressed as
- 18. Assignment Lesson 3.1 Page 123 Exercises: 1 41, 63 65 odd