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Lecture Notes 11: Differentiability, Tangent Line and Linearization I Instructor: Anatoliy Swishchuk Department of Mathematics & Statistics University of Calgary, Calgary, AB, Canada MATH 265 ’University Calculus I’ L01 Winter 2017 Outline of Lecture 1. Short Introduction 2. Differentiability 3. Tangent Line 4. Linearization Short Introduction: Two main Problems in Calculus In calculus, there are two main (or fundamental) problems: 1) the problem of slopes, and 2) the problem of areas. 1) The problem of slopes: finding the slope of (the tangent line to) a given curve at a given point on the curve; the solution of this problem is the subject of differential calculus (see Chapters 2-4 of our textbook). 2) The problem of areas: finding the area of a plane region bounded by curves and straight lines; the solution of this problem is subject of integral calculus (see Chapters 5-7 of our textbook). Usually we use the following notations: C for the graph of continuous function y = f (x); P for the point (x0, y0) on C, such that y0 = f (x0). Differentiability A function f (x) is differentiable at point a if f 0(a) exists: f (a + h) − f (a) 0 . f (a) = lim x→a h If f is differentiable at a, then f is continuous at a. Tangent Line to a General Curve We’ll consider the first problem that deals with finding a straight line L that is tangent to a curve C at a point P. Tangent Line to a Circle (with a center) A tangent line meets the circle at any point, the circle lies on only one side of the line, the tangent is perpendicular to the radius Tangent Line to a Curve (no center) a) L meets C only at P but is not tangent to C b) L meets C at several points but is tangent to C at P c) L is tangent to C at P but crosses C at P d) Many lines meet C only at P but none of them is tangent to C at P Secant Line If Q is a point on C different from P and Q is called a secant line to through P and Q, then we call it line notation for the coordinates of Q : P, then the line through the curve C. If line goes P Q. We use the following Q = (x0 + h, f (x0 + h)). Short Introduction: Slope, Difference Quotient The slope of the line P Q is: f (x0 + h) − f (x0) . (1) h This expression (1) is called the Newton quotient or difference quotient for f at x0. Definition of Nonvertival Tangent Lines If f continuous at x = x0 and f (x0 + h) − f (x0) lim = m = f 0(x0) h→0 h exists, then the straight line having slope m and passing through the point P = (x0, f (x0)) is called the tangent line (or simply the tangent) to the graph of y = f (x) at P. The equation of this tangent line is y = m(x − x0) + y0, where y = f (x), y0 = f (x0). (1) Example: An Equation of the Tangent Lines Find an equation of the tangent line to the curve y = x2 at the point (1, 1). Sol. Here, f (x) = x2, x0 = 1, y0 = f (1) = 1. The slope is: f (1+h)−f (1) (1+h)2 )−1 0 f (x0) = limh→0 = limh→0 h h 2 2 −1 2h+h = lim = limh→0 1+2h+h h→0 h h = limh→0(2 + h) = 2. According to (1) (see previous slide), we have y = 2(x − 1) + 1 or y = 2x − 1. Example: An Equation of the Tangent Lines (Figure) Definition of Vertical Tangent Lines If f continuous at P = (x0, y0), where y0 = f (x0), and if either f (x0 + h) − f (x0) f (x0 + h) − f (x0) = +∞ or lim = −∞, h→0 h→0 h h then the vertical line x = x0 is tangent to the graph y = f (x) at P. If the limit of the difference quotient fails to exist in any other way than by being +∞ or −∞, the graph y = f (x) has no tangent line at P. lim Vertical Tangent Lines: Example Here f (x) = x1/3, and slope approaches the infinity: 1/3 (0) h limh→0 f (0+h)−f = lim h→0 h h 1 = +∞. = limh→0 2/3 h Example 1: No Tangent Line (0) h2/3 = 1 has no limit Here: f (x) = x2/3 and f (0+h)−f = h h h1/3 (right is +∞ and left is −∞- cusp situation-infinitely sharp point). Example 2: No Tangent Line Here: y = |x| and there is no unique limit: |0 + h| − |0| |h| = = sgnh = h h ( 1, −1, if if h>0 h < 0. The Slope of a Curve: Definition The slope of a curve C at a point P is the slope of the tangent line to C at P if such a tangent line exists. The slope of the graph of y = f (x) at the point x0 is f (x0 + h) − f (x0) . h→0 h lim The Slope of a Curve: Example x Find the slope of the curve y = 3x+2 at the point x = −2. Sol. If x = 2, then y = 1/2, then the slope is: f 0(−2) = lim h→0 −2+h 1 − 2 3(−2+h)+2 h −h 1 = . h→0 2h(−4 + 3h) 8 = lim Normals to a Curve: Definition If a curve C has a tangent line L at point P, then the straight line N through P perpendicular to L is called the normal to C at P. The slope of N is the negative reciprocal of the slope of L : slope of the normal = − slope of 1 the tangent . Normals to a Curve: Example Find equations of the straight √ lines that are tangent and normal to the curve y = x at the point (4, 2). 1 . The tangent line is: Sol. Here, f 0(4) = 4 1 x (x − 4) + 2 or y = 1 + , 4 4 and the normal has slope −4, and equation y= y = −4(x − 4) + 2 or y = 18 − 4x. Normals to a Curve: Example with Figure Linearization Idea of Linearization: how to approximate some values of f (x)? We use the first derivative as an application of the tangent line to approximate f. The linear approximation of f near the point a has the following form: L(x) ≈ f 0(a)(x − a) + f (a). References 1) Calculus: Early Transcendental, 2016, An Open Text, by David Guichard: https : //lalg1.lyryx.com/textbooks/CALCU LU S 1/ucalgary/winter2016/math265/Guichard − Calculus − EarlyT rans − U of Calgary − M AT H265 − W 16.pdf 2) Optional Textbook: Essential Calculus, Early Transcendental, 2013, by J. Stewart, 2nd edition, Brooks/Cole