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1. The function h(t) = 400 – 162 gives the height of the supply bundle (in feet) t seconds after it is dropped from a height of 400 feet. a. Find the derivative of this function at the point (3, 256). b. What does your answer tell you about the speed at which the supply bundle is falling?

1. The function h(t) = 400 – 162 gives the height of the supply bundle (in feet) t seconds after it is dropped from a height of 400 feet. a. Find the derivative of this function at the point (3, 256). b. What does your answer tell you about the speed at which the supply bundle is falling?

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**MRN11 CC HW#13: Speeds, Rates, and Derivatives** **Part A. Meaningful Math** The derivative of a function at a point is one of the basic concepts of calculus. If a function is defined by the equation \( y = f(x) \) and \((a, b)\) is a point on the graph, then the *derivative* of \(f\) at \((a, b)\) can be thought of in at least two ways: - It is the slope of the line that is tangent to the curve at \((a, b)\). - It is the instantaneous rate at which the \( y \)-value of the function is changing as the \( x \)-value increases through \( x = a \). We often call this the derivative at \( x = a \) rather than the derivative at \((a, b)\). In this activity, you will work with this new idea in connection with some familiar situations. Keep in mind that you can find derivatives by using smaller and smaller intervals around a particular value of \(a\). **1.** The function \( h(t) = 400 - 16t^2 \) gives the height of the supply bundle (in feet) \( t \) seconds after it is dropped from a height of 400 feet. a. Find the derivative of this function at the point (3, 256). b. What does your answer tell you about the speed at which the supply bundle is falling? **2.** The function \( A(t) = \pi (70 + 6t)^2 \) gives the area of the oil slick (in square meters) \( t \) hours after Lindsay first spots it. a. Find the rate at which the area is growing exactly one hour after Lindsay first saw it. b. Express your answer as a derivative. **Image Description:** The image contains a photograph showing two jockeys on horseback in mid-race. This image serves as a visual metaphor for speed and rates, paralleling the mathematical concepts discussed.
Transcribed Image Text:**MRN11 CC HW#13: Speeds, Rates, and Derivatives** **Part A. Meaningful Math** The derivative of a function at a point is one of the basic concepts of calculus. If a function is defined by the equation \( y = f(x) \) and \((a, b)\) is a point on the graph, then the *derivative* of \(f\) at \((a, b)\) can be thought of in at least two ways: - It is the slope of the line that is tangent to the curve at \((a, b)\). - It is the instantaneous rate at which the \( y \)-value of the function is changing as the \( x \)-value increases through \( x = a \). We often call this the derivative at \( x = a \) rather than the derivative at \((a, b)\). In this activity, you will work with this new idea in connection with some familiar situations. Keep in mind that you can find derivatives by using smaller and smaller intervals around a particular value of \(a\). **1.** The function \( h(t) = 400 - 16t^2 \) gives the height of the supply bundle (in feet) \( t \) seconds after it is dropped from a height of 400 feet. a. Find the derivative of this function at the point (3, 256). b. What does your answer tell you about the speed at which the supply bundle is falling? **2.** The function \( A(t) = \pi (70 + 6t)^2 \) gives the area of the oil slick (in square meters) \( t \) hours after Lindsay first spots it. a. Find the rate at which the area is growing exactly one hour after Lindsay first saw it. b. Express your answer as a derivative. **Image Description:** The image contains a photograph showing two jockeys on horseback in mid-race. This image serves as a visual metaphor for speed and rates, paralleling the mathematical concepts discussed.
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