Dr. Frank J. Hayden Secondary SchoolMATH 4UVHS unit 9 done

Derivatives of Exponential and Logarithmic Functions Unit Assignment Note that derivatives must be fully simplified for full marks on this assignment. 1. Find the derivatives of each of these functions: (2 marks each) ...

Derivatives of Exponential and Logarithmic Functions Unit Assignment Note that derivatives must be fully simplified for full marks on this assignment. 1. Find the derivatives of each of these functions: (2 marks each) 1. y=ln (4 x2+4) 2. y=2 x4e5 x 3. y=25 x+7 (ln (5 x+1)) 4. y=4 x3e5 x+x4 2. Use the process of implicit differentiation to find d yd x given that x3 eyy ex=0. (4 marks) Implicit differentiation formula: This study source was downloaded by 100000793680026 from CourseHero.com on 06-28-2021 12:02:22 GMT -05:00 https://www.coursehero.com/file/96977903/VHS-unit-9-donepdf/ This study resource was shared via CourseHero.com3. Use curve sketching methods, sketch the graph of the function given by the equation y=x2ex. Make sure that you include all steps, charts, and derivations details. (10 marks) Therefore, the function will have local minima at(0,0) and a local maxima at (2, 4/e^2) Therefore, y=0 is the horizontal asymptote. This study source was downloaded by 100000793680026 from CourseHero.com on 06-28-2021 12:02:22 GMT -05:00 https://www.coursehero.com/file/96977903/VHS-unit-9-donepdf/ This study resource was shared via CourseHero.com4. Use logarithmic differentiation to find the derivative of the function y=(2 x+5)3 (5 x2-2 x)42 x+5. (6 marks) First we must take the logarithm of both sides. Now we must take the derivative of both sides. This study source was downloaded by 100000793680026 from CourseHero.com on 06-28-2021 12:02:22 GMT -05:00 https://www.coursehero.com/file/96977903/VHS-unit-9-donepdf/ This study resource was shared via CourseHero.com5. The population (P) of an island y years after colonisation is given by the function P=2501+4 e-0.01 y. 1. What was the initial population of the island? (2 marks) 2. How long did it take before the island had a population of 150? (2 marks) 3. After how many years was the population growing the fastest? (3 marks) Population growth will be calculated using first derivative. Population growth is fastest when using the second derivative. This study source was downloaded by 100000793680026 from CourseHero.com on 06-28-2021 12:02:22 GMT -05:00 https://www.coursehero.com/file/96977903/VHS-unit-9-donepdf/ This study resource was shared via CourseHero.com4. Using curve sketching methods, sketch the graph of the function. Make sure that you include all steps, charts, and derivations details. (10 marks) This study source was downloaded by 100000793680026 from CourseHero.com on 06-28-2021 12:02:22 GMT -05:00 https://www.coursehero.com/file/96977903/VHS-unit-9-donepdf/ This study resource was shared via CourseHero.comNow we must calculate the point of inflection. if f’’(x) > 0 , than f(x) will be concave up if f’’(x) < 0, than f(x) will be concave down 5. Give a possible explanation for the shape of the curve. (3 marks) This study source was downloaded by 100000793680026 from CourseHero.com on 06-28-2021 12:02:22 GMT -05:00 https://www.coursehero.com/file/96977903/VHS-unit-9-donepdf/ This study resource was shared via CourseHero.comGiven the from the graph shown, it is an exponentially increasing function that reaches its population limit of 250. This is most likely the carrying capacity of the population. This means, the island cannot hold any more people meaning it is a very small island that can only fit 250 people and if you are to exceed that limit their will not be enough room or resources for everyone. This is why the curve stops and shoots straight up. The curve goes through the point 50 which I think is telling us that 50 people is the point where the island becomes useful. Less than 50 people is not enough for the island. This study source was downloaded by 100000793680026 from CourseHero.com on 06-28-2021 12:02:22 GMT -05:00 https://www.coursehero.com/file/96977903/VHS-unit-9-donepdf/ This study resource was shared via CourseHero.com Powered by TCPDF (www.tcpdf.org)