Textbook Question
21–30. Derivatives
a. Use limits to find the derivative function f' for the following functions f.
f(t) = 3t⁴; a= -2, 2
2. Intro to Derivatives
Derivatives as Functions
6:23 minutesProblem 3.1.64a
Textbook Question
The following table gives the distance f(t) fallen by a smoke jumper seconds after she opens her chute. <IMAGE>
a. Use the forward difference quotient with ℎ = 0.5 to estimate the velocity of the smoke jumper at t=2 seconds.
Verified step by step guidance1
Step 1: Understand the forward difference quotient. The forward difference quotient is a method to estimate the derivative of a function at a certain point. It is given by the formula: \( f'(t) \approx \frac{f(t+h) - f(t)}{h} \), where \( h \) is a small increment.
Step 2: Identify the values needed from the table. You need the distance \( f(t) \) at \( t = 2 \) seconds and \( f(t+h) \) at \( t = 2.5 \) seconds, since \( h = 0.5 \).
Step 3: Substitute the values into the forward difference quotient formula. Use the values from the table for \( f(2) \) and \( f(2.5) \) to calculate the approximate velocity.
Step 4: Calculate the difference in distances. Find \( f(2.5) - f(2) \) to determine how much the distance has changed over the interval from \( t = 2 \) to \( t = 2.5 \).
Step 5: Divide the difference by \( h = 0.5 \). This will give you the estimated velocity of the smoke jumper at \( t = 2 \) seconds.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Forward Difference Quotient
The forward difference quotient is a method used to approximate the derivative of a function at a specific point. It is calculated using the formula (f(t + h) - f(t)) / h, where h is a small increment. In this context, it helps estimate the velocity of the smoke jumper by determining the rate of change of distance with respect to time.
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Velocity
Velocity is defined as the rate of change of an object's position with respect to time. In calculus, it is represented as the derivative of the position function, f(t), with respect to time, t. Understanding velocity is crucial for analyzing motion, as it provides insight into how fast an object is moving and in which direction.
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Estimation Techniques in Calculus
Estimation techniques in calculus, such as using difference quotients, allow for approximating values when exact calculations are difficult or impossible. These techniques are particularly useful in real-world applications, such as physics, where one may need to estimate rates of change based on discrete data points, like the distance fallen by the smoke jumper.
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