Plug t = 2 and t = 3 into the position equation to calculate the height of the object at the boundaries of the indicated interval to generate two ordered pair: (2, 1478) and (3, 1398). (b) Average velocity is the slope of the secant line, rather than the slope of the tangent line. The vecocity equation v( t) is the derivative of the position equation: v( t) = –32 t. Thus, the position equation is s( t) = –16 t2 + 1,542. Note that this position equation represents the height in feet of the object t seconds after it is released. (a) The position function for a projectile is s( t) = –16 t2 + v0 t + h0, where v0 represents the initial velocity of the object (in this case 0) and h0 represents the initial height of the object (in this case 1,542 feet). (e) If the object were to hit a six-foot-tall man squarely on the top of the head as he (unluckily) passed beneath, how fast would the object be moving at the moment of impact? Report your answer accurate to the thousandths place. (d) How many seconds does it take the object to hit the ground? Report your answer accurate to the thousandths place.
(c) Compute the velocity of the object 1, 2, and 3 seconds after it is released. (b) Calculate the average velocity of the object over the interval t = 2 and t = 3 seconds. (a) Construct the position and velocity equations for the object in terms of t, where t represents the number of seconds that have elapsed since the object was released. An object is dropped from the second-highest floor of the Sears Tower, 1542 feet off of the ground. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License. We recommend using aĪuthors: Gilbert Strang, Edwin “Jed” Herman Use the information below to generate a citation. Then you must include on every digital page view the following attribution: If you are redistributing all or part of this book in a digital format, Then you must include on every physical page the following attribution:
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Want to cite, share, or modify this book? This book uses theĬreative Commons Attribution-NonCommercial-ShareAlike License If f ( x ) f ( x ) is a function defined on an interval, , then the amount of change of f ( x ) f ( x ) over the interval is the change in the y y values of the function over that interval and is given by One application for derivatives is to estimate an unknown value of a function at a point by using a known value of a function at some given point together with its rate of change at the given point. These applications include acceleration and velocity in physics, population growth rates in biology, and marginal functions in economics. In this section we look at some applications of the derivative by focusing on the interpretation of the derivative as the rate of change of a function. 3.4.5 Use derivatives to calculate marginal cost and revenue in a business situation.3.4.4 Predict the future population from the present value and the population growth rate.3.4.3 Apply rates of change to displacement, velocity, and acceleration of an object moving along a straight line.3.4.2 Calculate the average rate of change and explain how it differs from the instantaneous rate of change.3.4.1 Determine a new value of a quantity from the old value and the amount of change.
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