Explanation Derivative is a way of finding the instantaneous rate of change. Derivative of a function can be represented as: f ' ( x ) = d f d x (Derivative of the function f with respect to the variable x ) Derivative of a function can be determined by the slope of the tangent line. We know that lim x → a f ( x ) = A means | f ( x ) − A | becomes arbitrarily small when x is sufficiently close to a , then the limit of f ( x ) as x → a is A . Given: The following graph of production of Crude oil Q ( t ) up to time t is given: Formula Used: The derivative of the function f ( x ) is given by the limit f ' ( x ) = lim h → 0 f ( x + h ) − f ( x ) h … ( 1 ) Calculation: (a) In order to sketch the derivative Q ' ( t ) at 1900 ≤ t ≤ 2150 , determine Q ' ( t ) for different values of t . For t = 1900 , use the equation (1) as below: Q ' ( t ) = lim t → 0 Q ( t + h ) − Q ( t ) h = lim t → 0 Q ( 1900 + 50 ) − Q ( 1900 ) 50 = lim t → 0 Q ( 1950 ) − Q ( 1900 ) 50 = 0.1 − 0 50 = 0.002 For t = 1950 , use the equation (1) as below: Q ' ( t ) = lim t → 0 Q ( t + h ) − Q ( t ) h = lim t → 0 Q ( 1950 + 50 ) − Q ( 1950 ) 50 = lim t → 0 Q ( 2000 ) − Q ( 1950 ) 50 = 0.7 − 0.1 50 = 0.012 For t = 2000 , use the equation (1) as below: Q ' ( t ) = lim t → 0 Q ( t + h ) − Q ( t ) h = lim t → 0 Q ( 2000 + 50 ) − Q ( 2000 ) 50 = lim t → 0 Q ( 2050 ) − Q ( 2000 ) 50 = 2...

3rd Edition

ISBN: 9781319030728

Author: Rogawski

Publisher: MAC HIGHER

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Question

Chapter 3.2, Problem 46E

To determine

(a) The sketch of the derivative $Q' (t)$ for $1900 \leq t \leq 2150$

(b) The year when $Q' (t)$ will be at its maximum value.

(c) $L = \lim_{t \to \infty} Q (t)$ and its interpretation

(d) The value of $\lim_{t \to \infty} Q' (t)$

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Chapter 3.2, Problem 45E

Chapter 3.2, Problem 47E

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Chapter 3 Solutions

Calculus: Early Transcendentals

Ch. 3.1 - Prob. 1PQ Ch. 3.1 - Prob. 2PQ Ch. 3.1 - Prob. 3PQ Ch. 3.1 - Prob. 4PQ Ch. 3.1 - Prob. 5PQ Ch. 3.1 - Prob. 1E Ch. 3.1 - Prob. 2E Ch. 3.1 - Prob. 3E Ch. 3.1 - Prob. 4E Ch. 3.1 - Prob. 5E

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(a) The sketch of the derivative Q ' ( t ) for 1900 ≤ t ≤ 2150 (b) The year when Q ' ( t ) will be at its maximum value. (c) L = lim t → ∞ Q ( t ) and its interpretation (d) The value of lim t → ∞ Q ' ( t )

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(a) The sketch of the derivative Q'(t)for 1900≤t≤2150 (b) The year when Q'(t) will be at its maximum value. (c) L=limt→∞Q(t) and its interpretation (d) The value of limt→∞Q'(t)

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(a) The sketch of the derivative $Q' (t)$ for $1900 \leq t \leq 2150$

(b) The year when $Q' (t)$ will be at its maximum value.

(c) $L = \lim_{t \to \infty} Q (t)$ and its interpretation

(d) The value of $\lim_{t \to \infty} Q' (t)$