MAE101 CAL v1 chapter 4 applications of derivatives

ĐẠI HỌC FPT CẦN THƠ Chapter APPLICATIONS OF DERIVATIVES (Page 341-485, 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 4.1 Related Rates 4.2 Linear Approximations and Differentials 4.3 Maxima and minima 4.4 The Mean value theorem 4.5 Derivatives and the shapes of Graphs 4.6 Limits at Infinity and Assymtotes 4.7 Applied Optimization Problems 4.9 Newton’s Method 4.10 Antiderivatives Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ APPLICATIONS OF DIFFERENTIATION 4.1 Related Rates In this section, we will learn: How to compute the rate of change of one quantity in terms of that of another quantity Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ RELATED RATES Example Air is being pumped into a spherical balloon so that its volume increases at a rate of 100 cm3/s How fast is the radius of the balloon increasing when the diameter is 50 cm? Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ Example The key thing to remember is that rates of change are derivatives – In this problem, the volume and the radius are both functions of the time t – The rate of increase of the volume with respect to time is the derivative dV / dt – The rate of increase of the radius is dr / dt Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ RELATED RATES Example To connect dV/dt and dr/dt, first we relate V and r by the formula for the volume of a sphere: V = r Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ Example To use the given information, we differentiate each side of the equation with respect to t To differentiate the right side, we need to use the Chain Rule: dV dV dr dr = = 4 r dt dr dt dt Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ RELATED RATES Example Now, we solve for the unknown quantity: dr dV = dt 4 dt If we put r = 25 and dV / dt = 100 in this equation, we obtain: dr 1 = 100 = dt 4 (25) 25 The radius of the balloon is increasing at the rate of 1/(25π) ≈ 0.0127 cm/s Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ Example A ladder 10 ft long rests against a vertical wall If the bottom of the ladder slides away from the wall at a rate of ft/s, how fast is the top of the ladder sliding down the wall when the bottom of the ladder is ft from the wall? Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ Example We first draw a diagram and label it as in the figure – Let x feet be the distance from the bottom of the ladder to the wall and y feet the distance from the top of the ladder to the ground – Note that x and y are both functions of t (time, measured in seconds) Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ CONVERGENCE If the numbers xn become closer and closer to r as n becomes large, then we say that the sequence converges to r and we write: lim xn = r n → Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ NEWTON’S METHOD Use Newton’s method to find places Example correct to eight decimal – First, we observe that finding is equivalent to finding the positive root of the equation x6 – = – So, we take f(x) = x6 – – Then, f’(x) = 6x5 Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ NEWTON’S METHOD Example So, Formula (Newton’s method) becomes: xn6 − xn +1 = xn − xn5 Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ Example NEWTON’S METHOD Choosing x1 = as the initial approximation, we obtain: x2 » 1.16666667 x3 » 1.12644368 x4 » 1.12249707 x5 » 1.12246205 x6 » 1.12246205 – As x5 and x6 agree to eight decimal places, we  1.12246205 conclude that Dr Tran Quoc Duy to eight decimal places Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ APPLICATIONS OF DIFFERENTIATION 4.10 Antiderivatives In this section, we will learn about: Antiderivatives and how they are useful in solving certain scientific problems Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ Definition A function F is called an antiderivative of f on an interval I if F’(x) = f(x) for all x in I Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ ANTIDERIVATIVES Theorem If F is an antiderivative of f on an interval I, the most general antiderivative of f on I is F(x) + C where C is an arbitrary constant Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ Let f(x)=tanx, and F(x) is an antiderivative of f(x) Evaluate f and tell whether F is increasing or decreasing at x= - (rad) Answer: Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ ANTIDERIVATIVE FORMULA Here, we list some particular antiderivatives Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ RECTILINEAR MOTION Example A particle moves in a straight line and has acceleration given by a(t) = 6t + Its initial velocity is v(0) = -6 cm/s and its initial displacement is s(0) = cm – Find its position function s(t) Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ A particle moves along the x-axis so that its velocity at time t is given by sin 2t Assuming it starts at the origin, where is it at t = π seconds? a b 3/2 c ½ d - 1/2 Answer: a Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ Let f(x)=4-3x for all x>=2 Select the correct one a is the local minimum value b is the local maximum value c -2 is the local minimum value d -2 is the maximum local value e is the absolute minimum value f -2 is the absolute maximum value g None of the above Answer: f Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ A piece of wire (dây kim loại) 10 m long is cut into two pieces One piece is bent into a square and the other is bent into an equilateral triangle (tam giác đều) How should the wire be cut for the square so that the total area (of the square and the triangle) enclosed is a minimum? Round the result to the nearest hundredth Answer: If x is the length of the square then the side of the triangle is 1/3 (10-4x) We find the minimum of x2+… Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ Dr Tran Quoc Duy Mathematics for Engineering ... 4. 1 Related Rates 4. 2 Linear Approximations and Differentials 4. 3 Maxima and minima 4. 4 The Mean value theorem 4. 5 Derivatives and the shapes of Graphs 4. 6 Limits at Infinity and Assymtotes 4. 7... Applied Optimization Problems 4. 9 Newton’s Method 4. 10 Antiderivatives Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ APPLICATIONS OF DIFFERENTIATION 4. 1 Related Rates In this section,... rate of change of one quantity in terms of that of another quantity Dr Tran Quoc Duy Mathematics for Engineering ĐẠI HỌC FPT CẦN THƠ RELATED RATES Example Air is being pumped into a spherical