Explanation Given information: We must show that: cosh ( x + y ) = cosh x cosh y + sinh x sinh y Formula used: Hyperbolic function of cos x and sin x : cosh x = e x + e − x 2 sinh x = e x − e − x 2 Proof: To show: cosh ( x + y ) = cosh x cosh y + sinh x sinh y Take Right hand side cosh ( x ) ⋅ cosh ( y ) + sinh ( x ) ⋅ sinh ( y ) = [ e x + e − x 2 ] ⋅ [ e y + e − y 2 ] + [ e x − e − x 2 ] [ e y − e − y 2 ] = [ ( e x + y + e x − y

4th Edition

ISBN: 9781319279813

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 1.6, Problem 45E

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cosh(x+y)=coshxcoshy+sinhxsinhy

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Chapter 1.6, Problem 44E

Chapter 1.6, Problem 46E

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Given lim x-4 f (x) = 1,limx-49 (x) = 10, and lim→-4 h (x) = -7 use the limit properties to find lim→-4 1 [2h (x) — h(x) + 7 f(x)] : - h(x)+7f(x) 3 O DNE

17. Suppose we know that the graph below is the graph of a solution to dy/dt = f(t). (a) How much of the slope field can you sketch from this information? [Hint: Note that the differential equation depends only on t.] (b) What can you say about the solu- tion with y(0) = 2? (For example, can you sketch the graph of this so- lution?) y(0) = 1 y AN

(b) Find the (instantaneous) rate of change of y at x = 5. In the previous part, we found the average rate of change for several intervals of decreasing size starting at x = 5. The instantaneous rate of change of fat x = 5 is the limit of the average rate of change over the interval [x, x + h] as h approaches 0. This is given by the derivative in the following limit. lim h→0 - f(x + h) − f(x) h The first step to find this limit is to compute f(x + h). Recall that this means replacing the input variable x with the expression x + h in the rule defining f. f(x + h) = (x + h)² - 5(x+ h) = 2xh+h2_ x² + 2xh + h² 5✔ - 5 )x - 5h Step 4 - The second step for finding the derivative of fat x is to find the difference f(x + h) − f(x). - f(x + h) f(x) = = (x² x² + 2xh + h² - ])- = 2x + h² - 5h ])x-5h) - (x² - 5x) = ]) (2x + h - 5) Macbook Pro

Chapter 1 Solutions

CALCULUS:EARLY TRANS.-SAPLINGPLUS

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Toprove: cosh ( x + y ) = cosh x cosh y + sinh x sinh y

Solution Summary: The author analyzes the hyperbolic function of mathrmcosx and

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Toprove:

cosh(x+y)=coshxcoshy+sinhxsinhy

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Chapter 1 Solutions

Toprove: cosh ( x + y ) = cosh x cosh y + sinh x sinh y

Solution Summary: The author analyzes the hyperbolic function of mathrmcosx and

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Toprove: cosh(x+y)=coshxcoshy+sinhxsinhy

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Chapter 1 Solutions

Toprove:

cosh(x+y)=coshxcoshy+sinhxsinhy