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Test 2. A tourist threw a pebble in the sea, causing water ripples. The shape formed were circles increasing in area. The formula for finding the area of a circle is given by A = mr?. 1. Find the average rate at which the area of a circle changes with r as the radius increases from 2 to 4 units. 2. Solve for the instantaneous rate at which the area changes with r, when r = 5.Homes derouton of dentolive may be used if we are tasked only to find the derivative of the junction) EXAMPLE 3: Let f(x) = 2r' + 3r - 1. Use the definition of derivative to find /(-1) Solution: F(x) = lim (Substituted ( r + Ah) to all variable) =lim (Expanded expressions) (Distributive Property of Multiplication) (Combined like terms IF IF IF IT (Factored out common monomial) - lim 4x + 24 +3 (Simplified form) = 4x + 2(0) +3 Evaluated the limit f(x) = 4r+] evaluate the function at /(-1), substitute -1 to the obtained derivative. "(-1)-4+3=40-1)+3--1EXAMPLE 4: Sepak Takraw is a sport associated to our traditional native sport "sipa." This sport requires agility, flexibility, and strength. It is a sport native to Southeast Asian countries such as the Philippines. A Sepak Takraw player kicked a ball at a certain height. Its height (in inches) from the ground at any time t (in seconds) is given by s(t) = 30 + 20t - 10t. Find: a. the height at which the ball was kicked b. the time when the ball hits the ground c. the average velocity of the ball on the interval [1,2] d. the instantaneous velocity at t = 1 and t = 2 e. the instantaneous velocity at any time to. Solutions: a. The height of the ball from the ground before it was kicked is denoted by t = 0. Thus, s(t) = 30 + 20t - 10t? = 30 + 20(0) - 10(0)? =(30 Therefore, the height of the ball from the ground before it was kicked is 30 inches. b. The ball is on the ground when the height s of the ball from the ground is 0. Thus, s(t) = 30 + 20t - 10t 0 = 30 + 20t - 10t2 0 = 10(3 - t)(1 + +) (Factored out the trinomial 3-1=0 1+t=0 (Solve for x) 1 3 t = -1 Since time is positive, disregard t = -1, then we choose t = 3 seconds. Therefore, the ball hit the ground after 3 seconds. c. The average velocity on the interval [1, 2] is defined by p = ()-f(t) where [t], t2) = [1,2] f(t,) = 30 + 20(1) - 10(1)= = 40 f(t2) = 30 + 20(2) - 10(2)? = 30 Substituting to the formula: f(t2) - f() 30-40 2-1 -10 Therefore, the average velocity of the ball is - 10 in/s. "(Negative velocity indicates that the direction of the ball is downward.)t-2 1-2 Therefore, the instantaneous velocity at time = 1 and time = 2 are 0 in/s and 20 in/s respectively. (t - 2) c. The instantaneous velocity at any time to. s (t) - s(to) (30 + 201 - 1013) - (30 + 20to - 1013) Jim = lim t - to t- to _ lim 104 207-1013-30-2010+ 106 (Distributive Property of Multiplication) = lim 201-1013-2060+1015 (Combined like terms) = lim 201-20t-1013+101 (Regrouped the terms) t-to = lim (Factored out common monomial) = lim (Factored out Difference of Two Squares) t-to = lim (1-) 20-10(8+to)] (Factored out common terms) t-to = lim [20 - 10(t + to)] (Simplified form) = 20 - 10(to + to) (Evaluated the limit) = 20 - 10(2to) (Simplified further) = 20 - 20t. (20- 20to) in/s (Affixed the unit) herefore, the instantaneous velocity function at any time to is (20 - 20to) in/s. Let Us Practice