Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
The Derivative and the Tangent Line Problem Lesson 3.1 Definition of Tan-gent 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) Slope of Line Tangent to a Curve • Approximated by secants – two points of intersection • •• • Let second point get closer and closer to desired View spreadsheet point of tangency simulation Animated Tangent Slope of Line Tangent to a Curve • Recall the concept of a limit from previous chapter • Use the limit in this context f ( x0 x) f ( x0 ) m lim x 0 x •• x Definition of a Tangent • Let Δx shrink from the left f ( x0 x) f ( x0 ) m lim x 0 x Definition of a Tangent • Let Δx shrink from the right f ( x0 x) f ( x0 ) m lim x 0 x The Slope Is a Limit • Consider f(x) = x3 x 0= 2 Find the tangent at f (2 x) f (2) m lim x 0 x (2 x)3 23 m lim x 0 x 2 3 8 12x 6(x) (x) 8 m lim x 0 x • Now finish … Animated Secant Line 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 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 Definition of Derivative • The derivative is the formula which gives the slope of the tangent line at any point x for f(x) f ( x0 x) f ( x0 ) f '( x) lim x 0 x • Note: the limit must exist – no hole – no jump – no pole – no sharp corner A derivative is a limit ! 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) 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 Using the Derivative • Consider that you are given the graph of the derivative … • What might the f `(x) slope of the original function look like? • Consider … – what do x-intercepts show? To actually find f(x), we – what do max and mins show?need a specific point it contains – f `(x) <0 or f `(x) > 0 means what? Derivative Notation • For the function y = f(x) • Derivative may be expressed as … f '( x) "f prime of x" dy "the derivative of y with respect to x" dx Assignment • Lesson 3.1 • Page 123 • Exercises: 1 – 41, 63 – 65 odd