mathematics in the modern world, Assignments of Mathematics

Download mathematics in the modern world and more Assignments Mathematics in PDF only on Docsity! THE DERIVATIVE y=f(x) y+∆y=f(x+∆x) , ∆y=f(x+∆x)-f(x) then we will form ∆𝑦 ∆𝑥 𝑎𝑛𝑑 𝑡ℎ𝑖𝑠 𝑟𝑎𝑡𝑖𝑜 𝑖𝑠 𝑐𝑎𝑙𝑙𝑒𝑑 𝑎 𝑠𝑙𝑜𝑝𝑒 or derivative of y with respect to x when ∆x approches 0 lim ∆𝑥 0 ∆𝑦 ∆𝑥 = 𝑑𝑦 𝑑𝑥 = lim ∆𝑥 0 𝑓(𝑥−∆𝑥)−𝑓(𝑥) ∆𝑥 Other symbol for derivative y’, f’(x), Dxy , 𝑑 𝑑𝑥 𝑓(𝑥) To obtain of any function the general process is as follows: 1. Replace x by x+∆x and y by y+∆y 2. Subtract y by both side to obtain a formula for ∆y in terms of x and ∆x 3. By some suitable transformation, throw the right member into form w/c contains ∆x as explicity factor. 4. Divide through by ∆x 5. Determine the limit as ∆x approches 0 Example: y=x 1. Y+∆y=x+∆x 2. ∆y=x+∆x-x 3. ∆y=∆x 4. ∆𝑦 ∆𝑥 = ∆𝑥 ∆𝑥 =1 5. lim ∆𝑥 0 1 = 1 Example : Differentiate the function y=𝑥3 − 2𝑥 1. y+∆y=(𝑥 + ∆𝑥)3 − 2(𝑥 − ∆𝑥) 2. ∆y=(𝑥 + ∆𝑥)3 − 2(𝑥 − ∆𝑥) - 𝑥3 + 2𝑥 3. ∆y= 𝑥3 + 3𝑥2∆𝑥 + 3∆𝑥2𝑥 + ∆𝑥3 − 2𝑥 − 2∆𝑥 − 𝑥3 + 2𝑥 ∆y= 3𝑥2∆𝑥 + 3∆𝑥2𝑥 + ∆𝑥3 − 2∆𝑥 4. ∆𝑦 ∆𝑥 = 3𝑥2 + 3𝑥∆𝑥 + ∆𝑥2 − 2 5. y’=3𝑥2 − 2 Example: Differentiate y=4𝑥2 − 3𝑥 − 2 1. y+∆y=4(𝑥 + ∆𝑥)2 − 3(𝑥 + ∆𝑥) − 2 2. ∆y= 4(𝑥 + ∆𝑥)2 − 3(𝑥 + ∆𝑥) − 2 - 4𝑥2 + 3𝑥 + 2 3. ∆y=4𝑥2 − 8𝑥∆𝑥 + 4∆𝑥2 − 3𝑥 + 3∆𝑥 − 4𝑥2 + 3𝑥 ∆y=8x∆x+4∆𝑥2 + 3∆𝑥 4. ∆𝑦 ∆𝑥 = 8𝑥 + 4∆𝑥 + 3 5. y’= 8x+3 Example: Differentiate y=√𝑥 1. y+∆y=√𝑥 + ∆𝑥 2. ∆y=√𝑥 + ∆𝑥-√𝑥 3. ∆y=√𝑥 + ∆𝑥-√𝑥 . √𝑥+∆𝑥+√𝑥 √𝑥+∆𝑥+√𝑥 = (√𝑥+∆𝑥) 2 −(√𝑥) 2 √𝑥+∆𝑥+√𝑥 = 𝑥+∆𝑥−𝑥 √𝑥+∆𝑥+√𝑥 = ∆𝑥 √𝑥+∆𝑥+√𝑥 4. ∆𝑦 ∆𝑥 = ∆𝑥 √𝑥+∆𝑥+√𝑥 . 1 ∆𝑥 = 1 √𝑥+∆𝑥+√𝑥 5. y’= 1 √𝑥+√𝑥 = 1 2√𝑥 Example: y=√𝑥 + 2 1. y+∆y= √𝑥 + ∆𝑥 + 2 2. ∆y= √𝑥 + ∆𝑥 + 2-√𝑥 + 2