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For Problems 1-2, find the derivative of the given fu appear. Show enough work so that it's clear that you final answer. 1. f(x) = (x²+x+1)eª 3. Find the equation of the tangent line to the par= 4. A dam having a width of 500 m is holding back a

For Problems 1-2, find the derivative of the given fu appear. Show enough work so that it's clear that you final answer. 1. f(x) = (x²+x+1)eª 3. Find the equation of the tangent line to the par= 4. A dam having a width of 500 m is holding back a

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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**Derivative Problems and Tangent Lines** For Problems 1–2, find the derivative of the given function. Assume that \( a, b, \) and \( c \) are constant wherever they appear. Show enough work so that it’s clear that you, and not your calculator, took the derivative, and circle your final answer. 1. \( f(x) = (x^2 + x + 1) e^x \) 2. \( f(x) = \frac{2 + \sin x}{x + \cos x} \) 3. Find the equation of the tangent line to the parabola \( y = 2x^2 - x + 3 \) at \( x = 2 \). Circle your final answer. 4. A dam having a width of 500 m is holding back a column of water of height \( H \) meters. The total force on the dam is given by \( F(H) = 2450H^2 \) kilonewtons. a) Find \( F'(50) \), and include the correct units with your answer. b) Give a complete sentence interpretation of \( F'(50) \) in the context of this problem. **Diagram Explanation:** The diagram on the right illustrates a dam holding back water. The dam is shown in cross-section, with water pressing against the dam face. The key features of the diagram include: - The width of the dam (\(500\) meters) is indicated by a horizontal arrow. - The height of the water column (\(H\) meters) is indicated by a vertical arrow, illustrating the depth of the water from the top to the bottom of the dam.
Transcribed Image Text:**Derivative Problems and Tangent Lines** For Problems 1–2, find the derivative of the given function. Assume that \( a, b, \) and \( c \) are constant wherever they appear. Show enough work so that it’s clear that you, and not your calculator, took the derivative, and circle your final answer. 1. \( f(x) = (x^2 + x + 1) e^x \) 2. \( f(x) = \frac{2 + \sin x}{x + \cos x} \) 3. Find the equation of the tangent line to the parabola \( y = 2x^2 - x + 3 \) at \( x = 2 \). Circle your final answer. 4. A dam having a width of 500 m is holding back a column of water of height \( H \) meters. The total force on the dam is given by \( F(H) = 2450H^2 \) kilonewtons. a) Find \( F'(50) \), and include the correct units with your answer. b) Give a complete sentence interpretation of \( F'(50) \) in the context of this problem. **Diagram Explanation:** The diagram on the right illustrates a dam holding back water. The dam is shown in cross-section, with water pressing against the dam face. The key features of the diagram include: - The width of the dam (\(500\) meters) is indicated by a horizontal arrow. - The height of the water column (\(H\) meters) is indicated by a vertical arrow, illustrating the depth of the water from the top to the bottom of the dam.
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