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Analytical Example:

Part of the usability of derivatives comes from its real world application. Derivatives can be used when calculating position, velocity, and acceleration from f(x), f(x), and f(x) respectively. By being able to translate between each graph, we can accurately describe a position function and its rates. How else would we be able to tell Harry Potters velocity as he falls from his broom after encountering a mob of dementors during his quidditch match before Dumbledore stops him? Or how quickly he must accelerate to catch the golden snitch? Or how fast a bludger is heading towards the Weasley twins? Furthermore, we can use the derivative to find maximums and minimums, often useful in business. For example, the goblins of Gringotts often use derivatives to maximize their profits.

Given that f(x)=x(3x+2)2, find the first and second derivatives.

First , you must recognize that this incorporates the chain rule with in the product rule. This means that we will use the product rule first and the chain rule will be used as a result of the product rule.

As you can see, we used the basic formula, first*dsecond + second*dfirst, but when you are taking the derivative of (3x+2)2 you must use the chain rule and bring the 2 out and multiply by the derivative of what's inside the parenthesis.

Dont forget to simplify!

Remember that when you are asked to take simple derivatives, you are lowering the variable by one degree and multiplying its constant by its previous power.

F(x)= x(3x+2)2

F(x)= 6x(3x+2)+(3x+2)2

F(x)=18x2+12x+9x2+12x+4

F(x)=27x2+24x+4

F(x)=54x+24

Neville Longbottom is flying on a broom that is losing control. His position is given by the function f(X)=x3-6x2+11x+7 where 1 x 3. Find the maximum distance Neville reached from his origin. Calculators at the ready!

a) 1.423 b) 13.385 c) 12.615 d) 3.838 e) 2.577 To approach this problem, we first have to remember that a maximum can be determined when the derivative of the function from positive to negative or the original functions endpoints. Take the derivative of the function and plug it into Y1 of your calculator. Y1 = 4x3-12x+11Graph the derivative in your calculator and use the 2nd Calc function to find the values where the derivative changes from positive to negative and the end points. These are going to be the possible time where Neville is at his maximum. Make sure to store these values in your calculator. Plug these times back into the original position function of x and compare the distances to find Nevilles maximum value.

As you can see, 13.385 is the maximum value Neville is from his origin, making (b) the correct answer. xf(x)071.42313.385313

Letter A is incorrect because it is simply the time at which the derivative changes from positive to negative, not the distance which Neville has traveled.Letter B is the correct answer.Letter C is incorrect because it is the distance if you had found the value of the time the derivative changes from negative to positive and plugged it into Nevilles position function, which represents his minimum distance from the origin, not the maximum distance.Letter D is a random answer. We dearly hope you did not end up with answer D.Letter E is the time when the Nevilles velocity changes from negative to positive, making it the time Neville would be the minimum distance from the origin and the incorrect answer.

Harry Potter and Ron Weasley have missed the train to Hogwarts. They decide to take Mr. Weasleys flying car to catch up to the train. Far in the distance, they see the train and Harry notices that the train is 5 km away from the station. He also calculates that it is traveling .5km/sec and decelerating at a rate of 1km/sec2. Using this information, calculate the position of the train the next second. a.) 5.5 km b.) 5.7 km c.) 6 km d.) 5.4 km e.) 5 km

If we use the fact that y=5, y=.5, and y=-1, we can come to the conclusion that D (5.4) is the right answer!

-Answer choice A is not correct because that would mean that the train is traveling at a constant rate, which doesnt take into account the fact that the train is decelerating. -Answer choices B and C are not correct because the position of the train is increasing with a decreasing rate, which is given by the positive velocity and the negative acceleration, so the trains position cant possibly be more than 5.5km.-Answer choice E is not correct because that would mean that the train has stopped and has remained at its original position of 5 km.

t012345678P(t)05.47.910.912.314.314.915.716.2Harry Potter and Ron Weasley have been making an polyjuice potion in their potions class while Professor Snape was not looking. The amount of potion theyve made at time t, 0 < t < 8, is given by a differentiable function P, where t is measured in minutes. Selected values of P(t), measured in ounces, are given in the table above.

a.) Is there a time t, 3 < x < 7, where P(t)=2? Justify your answer.

AP Level Free Response QuestionAP Level Free Response Question: Answer For this problem, we have to utilize the mean value theorem: the average slope (or secant line) over an interval must equal the instantaneous rate of change at one point on that interval.

Using this method, we find that there is a time t, between 4 < t < 5, that the rate at which the potion is being made is 2 ounces/min. Graphical Example

Given the graph of f(x) below, at what time does f(x) have a relative minimum? 1325476a) 1, 4, 6 b) 2, 5, 7 c) 2, 7 d) 3, 5 e)3Since we are looking for relative extrema of f(x) using f(x) we are looking for places where f(x) changes signs. This eliminates numbers 1, 4, and 6. Because we are looking for minimums in particular, we want places where f(x) changes from negative to positive, because this will be time where the slopes of f(x) changes from negative to positive, leaving a minimum. This eliminates number 3, in which f(x) changes from positive to negative.Although f(x) looks like its about to change signs at 5, it merely kisses the x axis, also eliminating from possible answer choices. Therefore, f(x) has relative minimum at 2 and 7 where f(x) changes from negative to positive. C is the correct answer.

Harry Potter survived through these exercises! Lets see if you can too! O.W.L.Ordinary Wizarding Level Examination(Exercises)Analytical Exercises

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AP Multiple Choice Questions

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