Explanation Given information: The function is f ( t ) = sin t + cos t and the range of t is − 2 π ≤ t ≤ 2 π . Graph: To construct the graph, use the following steps: Step a: The given function is f ( t ) = sin t + cos t … ( 1 ) Differentiate the above function with respect to t , f ′ ( t ) = d d t ( sin t + cos t ) ⇒ f ′ ( t ) = cos t − sin t . … ( 2 ) Substitute f ′ ( t ) = 0 in the above equation ⇒ f ′ ( t ) = cos t − sin t = 0 ⇒ cos t = sin t When cos t = sin t , then t = ( 4 n + 1 ) π 4 where n is an integer. Therefore, for t ∈ [ − 2 π , 2 π ] f ′ ( t ) = 0 ⇒ t = π 4 , 5 π 4 , − 3 π 4 , − 7 π 4 . Therefore, the points at which f ′ ( t ) = 0 are t = π 4 , 5 π 4 , − 3 π 4 , − 7 π 4 . Step b: To check the concavity of f ( t ) , differentiate equation ( 2 ) with respect to t f ″ ( t ) = d d t ( cos t − sin t ) ⇒ f ″ ( t ) = − sin t − cos t … (3) If f ″ ( t 0 ) is positive, then the curve is concave up at t = t 0 and if f ″ ( t 0 ) is negative, then the curve is concave down at t = t 0 . So, at t = π 4 , f ″ ( π 4 ) = − sin ( π 4 ) − cos ( π 4 ) = − 1 2 − 1 2 = − 2 2 ⇒ f ″ ( π 4 ) = − 2 < 0 So, the curve is concave down at t = π 4 . At t = 5 π 4 f ″ ( 5 π 4 ) = − sin ( 5 π 4 ) − cos ( 5 π 4 ) = − ( − 1 2 ) − ( − 1 2 ) = 1 2 + 1 2 = 2 2 = 2 ⇒ f ″ ( 5 π 4 ) = 2 > 0 Therefore, the curve is concave up at t = 5 π 4 . At t = − 3 π 4 f ″ ( − 3 π 4 ) = − sin ( − 3 π 4 ) − cos ( − 3 π 4 ) = − ( − 1 2 ) − ( − 1 2 ) = 1 2 + 1 2 = 2 2 = 2 ⇒ f ″ ( − 3 π 4 ) = 2 > 0 So, the curve is concave up at t = − 3 π 4

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ISBN: 9780135901236

Author: Goldstein

Publisher: PEARSON

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Riemann Sum

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Riemann Integral

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Chapter 8, Problem 50RE

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To graph: The function f(t)=sint+cost for −2π≤t≤2π, by using the points t between −2π to 2π such that f′(t)=0, concavity of the function f(t) and the inflection points of the function f(t).

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Chapter 8, Problem 49RE

Chapter 8, Problem 51RE

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Ch. 8.1 - A right triangle has one angle of /3 radians. What...Ch. 8.1 - Prob. 2CYU Ch. 8.1 - Prob. 1E Ch. 8.1 - Prob. 2E Ch. 8.1 - Prob. 3E Ch. 8.1 - Prob. 4E Ch. 8.1 - Give the radian measure of each angle described.Ch. 8.1 - Prob. 6E Ch. 8.1 - Prob. 7E Ch. 8.1 - Prob. 8E

Ch. 8.1 - Prob. 9E Ch. 8.1 - Prob. 10E Ch. 8.1 - Prob. 11E Ch. 8.1 - Prob. 12E Ch. 8.1 - Prob. 13E Ch. 8.1 - Prob. 14E Ch. 8.1 - Prob. 15E Ch. 8.1 - Prob. 16E Ch. 8.1 - Prob. 17E Ch. 8.1 - Prob. 18E Ch. 8.2 - Find cost, where t is the radian measure of the...Ch. 8.2 - Prob. 2CYU Ch. 8.2 - Prob. 1E Ch. 8.2 - Prob. 2E Ch. 8.2 - Prob. 3E Ch. 8.2 - Prob. 4E Ch. 8.2 - In Exercises 112, give the values of sint and...Ch. 8.2 - Prob. 6E Ch. 8.2 - Prob. 7E Ch. 8.2 - Prob. 8E Ch. 8.2 - Prob. 9E Ch. 8.2 - Prob. 10E Ch. 8.2 - Prob. 11E Ch. 8.2 - Prob. 12E Ch. 8.2 - Prob. 13E Ch. 8.2 - Prob. 14E Ch. 8.2 - Prob. 15E Ch. 8.2 - Prob. 16E Ch. 8.2 - Prob. 17E Ch. 8.2 - Prob. 18E Ch. 8.2 - Prob. 19E Ch. 8.2 - Prob. 20E Ch. 8.2 - Prob. 21E Ch. 8.2 - Prob. 22E Ch. 8.2 - Prob. 23E Ch. 8.2 - Prob. 24E Ch. 8.2 - Prob. 25E Ch. 8.2 - Prob. 26E Ch. 8.2 - Prob. 27E Ch. 8.2 - Prob. 28E Ch. 8.2 - Prob. 29E Ch. 8.2 - Prob. 30E Ch. 8.2 - Prob. 31E Ch. 8.2 - Prob. 32E Ch. 8.2 - Prob. 33E Ch. 8.2 - Prob. 34E Ch. 8.2 - Prob. 35E Ch. 8.2 - Prob. 36E Ch. 8.2 - Prob. 37E Ch. 8.2 - Prob. 38E Ch. 8.2 - Prob. 39E Ch. 8.2 - Prob. 40E Ch. 8.2 - Prob. 41E Ch. 8.2 - In any given locality, the length of daylight...Ch. 8.3 - Differentiate y=2sin[t2+(/6)].Ch. 8.3 - Prob. 2CYU Ch. 8.3 - Prob. 1E Ch. 8.3 - Prob. 2E Ch. 8.3 - Prob. 3E Ch. 8.3 - Prob. 4E Ch. 8.3 - Differentiate (with respect to t or x): y=2cos3t Ch. 8.3 - Differentiate (with respect to t or x): y=sin3t3 Ch. 8.3 - Prob. 7E Ch. 8.3 - Differentiate (with respect to t or x): y=tcost Ch. 8.3 - Prob. 9E Ch. 8.3 - Prob. 10E Ch. 8.3 - Differentiate (with respect to t or x): y=cos3t Ch. 8.3 - Prob. 12E Ch. 8.3 - Prob. 13E Ch. 8.3 - Prob. 14E Ch. 8.3 - Prob. 15E Ch. 8.3 - Prob. 16E Ch. 8.3 - Prob. 17E Ch. 8.3 - Prob. 18E Ch. 8.3 - Prob. 19E Ch. 8.3 - Prob. 20E Ch. 8.3 - Prob. 21E Ch. 8.3 - Prob. 22E Ch. 8.3 - Prob. 23E Ch. 8.3 - Prob. 24E Ch. 8.3 - Prob. 25E Ch. 8.3 - Prob. 26E Ch. 8.3 - Prob. 27E Ch. 8.3 - Prob. 28E Ch. 8.3 - Prob. 29E Ch. 8.3 - Prob. 30E Ch. 8.3 - 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To graph: The function f ( t ) = sin t + cos t for − 2 π ≤ t ≤ 2 π , by using the points t between − 2 π to 2 π such that f ′ ( t ) = 0 , concavity of the function f ( t ) and the inflection points of the function f ( t ) .

Solution Summary: The author explains the function f(t) = mathrmsint+

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To graph: The function f(t)=sint+cost for −2π≤t≤2π, by using the points t between −2π to 2π such that f′(t)=0, concavity of the function f(t) and the inflection points of the function f(t).

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Chapter 8 Solutions