MAE101 CAL v1 chapter 3 derivatives

ĐẠI HỌC FPT CẦN THƠ Chapter DERIVATIVES (Page 213-319, Calculus Volume 1) Dr Tran Quoc Duy Email: duytq4@fpt.edu.vn Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ CONTENTS 3.1 Defining the Derivative 3.2 The Derivative as a Function 3.3 Differentiation Rules 3.4 Derivatives as Rates of Change 3.5 Derivatives of Trigonometric Functions 3.6 The Chain Rule 3.7 Derivatives of Inverse Functions 3.8 Implicit Differentiation 3.9 Derivatives of Exponential and Logarithmic Functions Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ DERIVATIVES 3.1 Defining the Derivatives In this section, we will learn: How the derivative can be interpreted as a rate of change in any of the sciences or engineering Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ THE TANGENT PROBLEM Example Find an equation of the tangent line to the parabola y = x2 at the point P(1,1) We will be able to find an equation of the tangent line as soon as we know its slope m x -1 = x -1 mPQ Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ THE TANGENT PROBLEM Example The slope of the tangent line is said to be the limit of the slopes of the secant lines lim mPQ = m Q®P x -1 lim =2 x ®1 x - The equation of the tangent line through (1, 1) as: y = 2x -1 Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ TANGENTS Definition The tangent line to the curve y = f(x) at the point P(a, f(a)) is the line through P with slope f ( x) - f (a) m = lim x®a x-a provided that this limit exists Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ THE VELOCITY PROBLEM Example Investigate the example of a falling ball ▪ Suppose that a ball is dropped from the upper observation deck of the CN Tower in Toronto, 450 m above the ground ▪ Find the velocity of the ball after seconds Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ THE VELOCITY PROBLEM Example If the distance fallen after t seconds is denoted by s(t) and measured in meters, then Galileo’s law is expressed by the following equation s(t) = 4.9t2 Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ THE VELOCITY PROBLEM change in position average velocity = time elapsed = s ( 5.1) − s ( ) = 49.49 m/s 0.1 Thus, the (instantaneous) velocity after s is: v = 49 m/s Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ VELOCITIES Definition We define the velocity (or instantaneous velocity) v(a) at time t = a to be the limit of these average velocities: f ( a + h) − f ( a ) v(a) = lim h →0 h Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ THE CHAIN RULE If g is differentiable at x and f is differentiable at g(x), the composite function F = f ◦ g is differentiable at x and F’ is given by the product: F’(x) = f’(g(x)) • g’(x) – In Leibniz notation, if y = f(u) and u = g(x) are both differentiable functions, then: dy dy du = dx du dx Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ Let f(x)=g(sin3x) Find f ’ in terms of g’ A 3cos3xg’(x) B 3cos3xg’(sin3x) C cos3xg’(sin3x) Answer: b Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ Suppose h(x)=f(g(x)) and f(2)=3, g(2)=1, g’(2)=1, f’(2)=2, f’(1)=5 Find h’(2) A B C D E Answer: e Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ DERIVATIVES 3.6 Implicit Differentiation Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ IMPLICIT DIFFERENTIATION The graphs of f and g are the upper and lower semicircles of the circle x2 + y2 = 25 Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ IMPLICIT DIFFERENTIATION METHOD Instead, we can use the method of implicit differentiation – This consists of differentiating both sides of the equation with respect to x and then solving the resulting equation for y’ Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ Example IMPLICIT DIFFERENTIATION a If x2 + y2 = 25, find dy dx b Find an equation of the tangent to the circle x2 + y2 = 25 at the point (3, 4) d d ( x + y ) = (25) dx dx d d (x ) + ( y ) = dx dx Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ Example a IMPLICIT DIFFERENTIATION Remembering that y is a function of x and using the Chain Rule, we have: d d dy dy (y ) = (y ) = 2y dx dy dx dx dy 2x + y =0 dx Then, we solve this equation for Dr Tran Quoc Duy dy dx : dy x =− dx y Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ IMPLICIT DIFFERENTIATION E g b—Solution At the point (3, 4) we have x = and y = So, dy = − dx Thus, an equation of the tangent to the circle at (3, 4) is: y – = – ¾(x – 3) or 3x + 4y = 25 Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ IMPLICIT DIFFERENTIATION Example Find y” if x4 + y4 = 16 Differentiating the equation implicitly with respect to x, we get 4x3 + 4y3y’ = Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ IMPLICIT DIFFERENTIATION E g 4—Equation Solving for y’ gives: x y' = - y Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ IMPLICIT DIFFERENTIATION Example To find y’’, we differentiate this expression for y’ using the Quotient Rule and remembering that y is a function of x: d  x3  y (d / dx)( x ) − x (d / dx)( y ) y '' =  −  = − dx  y  (y ) y  3x − x (3 y y ') =− y6 Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ Example IMPLICIT DIFFERENTIATION If we now substitute Equation into this expression, we get:  x  3 3x y − 3x y  −  y   y '' = − y6 3( x y + x ) 3x ( y + x ) =− =− 7 y y Dr Tran Quoc Duy 4 Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ IMPLICIT DIFFERENTIATION Example However, the values of x and y must satisfy the original equation x4 + y4 = 16 So, the answer simplifies to: 2 3x (16) x y '' = − = −48 7 y y Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ Dr Tran Quoc Duy Mathematics for Engineering ... CONTENTS 3. 1 Defining the Derivative 3. 2 The Derivative as a Function 3. 3 Differentiation Rules 3. 4 Derivatives as Rates of Change 3. 5 Derivatives of Trigonometric Functions 3. 6 The Chain Rule 3. 7 Derivatives. .. n dx Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ DERIVATIVES 3. 3 Differentiation Rules 3. 4 Derivatives of Trigonometric functions 3. 5 Derivatives of Exponential and Logarithmic functions... Functions 3. 8 Implicit Differentiation 3. 9 Derivatives of Exponential and Logarithmic Functions Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ DERIVATIVES 3. 1 Defining the Derivatives