Homework Answers
Using
above graph, it's clear that tanhx & sinhx increases in it's
domain.
Hence, option D is correct.
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Which hyperbolic functions are increasing on their domains? A. tanh x and cosh x are increasing...
Compose a module that implements the hyperbolic trigonometric functions based on the definitions sinh(x) = (e...
Compose a module that implements the hyperbolic trigonometric functions based on the definitions sinh(x) = (e – e ) / 2 and cosh(x) = (e + e ) / 2, with tanh(x), coth(x), sech(x), and csch(x) defined in a manner analogous to the standard trigonometric functions. In Python
The hyperbolic cosine and hyperbolic sine functions, f(x) cosh(x) and g(x) sinh(), are analogs of the trigonometric functions cos(x) and sin(z) and come up in many places in mathematics and its appli...
The hyperbolic cosine and hyperbolic sine functions, f(x) cosh(x) and g(x) sinh(), are analogs of the trigonometric functions cos(x) and sin(z) and come up in many places in mathematics and its applications. (The hyperbolic cosine, for example, describes the curve of a hanging cable, called a catenary.) They are defined by the conditions cosh(0)-l, sinh(O), (cosh())inh("), d(sinh()- csh) (a) Using only this information, find the Taylor polynomial approximation for cosh(x) at0 of COS degree n = 4. (b) Using only...
Recall three facts about hyperbolic cosine and hyperbolic sine functions: dr cosh(x) = sinh(x), di sinh(x)...
Recall three facts about hyperbolic cosine and hyperbolic sine functions: dr cosh(x) = sinh(x), di sinh(x) = cosh(x), cosh? (x) – sinh? (x) = 1. Compute the Wronskian of cosh(2) and sinh(2). Using your question 1's answer, cosh() and sinh(x) are linearly independent on R True False
' ' 3. The hyperbolic trigonometric functions are defined by sinh(t) , The following identities m...
' ' 3. The hyperbolic trigonometric functions are defined by sinh(t) , The following identities might be helpful in the following problem. You should convince yourself that they are true, but you are not required to write up proofs/derivations cosh2 1-sinh2に1, cosh2 1 + sinh2にcosh(21), cosh 1 -sinh 1-cosh 1 sinh 1, dl dl (a) For a real number t, define sinh-1に1n(t + Vt2 + 1) Show that sinh(t) is the unique eaber u such that sinhu t (b) Use...
5. Use a substitution and an integration by parts to find each of the following indef-...
5. Use a substitution and an integration by parts to find each of the following indef- inite integrals: (b) | (cos(a) sin(a) esas) de (a) / ( (32 – 7) sin(5x + 2)) de (c) / (e* cos(e=)) dt (d) dr 6. Spot the error in the following calculation: S() will use integration by parts with 1 We wish to compute dr. For this dv du 1 dar = 1. This gives us dr by parts we find dr =...
question 10 and 11 10. Solve the equation 5 cosh x – 3 sinh x =...
question 10 and 11
10. Solve the equation 5 cosh x – 3 sinh x = 5. 11. (a) Show that cosh (x - y) = cosh x cosh y - sinh x sinh y. (b) Show that for any real number x, cosh” x + sinh? x = cosh2x . Hence prove that cosh2x = 2 sinhạ x + 1 = 2 cosh? x - 1. 1+ tanh x (c) Show that 1- tanh x
In class we discussed the relationship between the hyperbolic functions and a hyperbola then showed that it is analogou...
In class we discussed the relationship between the hyperbolic functions and a hyperbola then showed that it is analogous to that of the trigonometric functions and a circle a. Derive an analogue to the Pythagorean Identities (cos2 x + sin2 x 1, etc. ) for the hyperbolic functions hint: Which hyperbola and which circle? (this will give you the relationship between cosh x and sinh x and the others are then easily found as they were in the case of...
Problem 7. (20 points) We consider the function tanh(z) sinh(z) tanh(z)=cosh(z) where , For any integer...
Problem 7. (20 points) We consider the function tanh(z) sinh(z) tanh(z)=cosh(z) where , For any integer 0, we denote by Qe the positively oriented square whose edges lie along the lines z-t(k+1) π and y = ± (k+)π -(km 4p. (a) Show that for any z z + iy e C, |cosh(z)12-sinh2(x) + cos2(y). 2p (b) Recall that tanh is analytic at the origin and that tanh () 1 - tanh2(). Compute the tanh(z) limit l := lim (Problem 7...
10. Find the first derivative with respect to the independent variable for the following functions: (a)...
10. Find the first derivative with respect to the independent variable for the following functions: (a) f(x) = sinh(4x) (b) g(t) = cosh(t) sinh(t). 1 - cosh(r) (c) h(r) = 1 + cosh(r) (d) F(X) = tanh(e). (e) y = sinh--(VT). (Hint: Apply implicit differentiation to sinh(y) = VT.) 11. Use appropriate hyperbolic function substitutions to evaluate the following indef- inite integrals: dc (a) /+) Ke-10) (e)/() dx (b) dar James G., Modern Engineering Mathematics (5th ed.) 2015.: Exercise set...
If a body of mass m falling from rest under the action of gravity encounters an...
If a body of mass m falling from rest under the action of gravity encounters an air resistance proportional to the square of the velocity, then the body's velocity t sec into dv the fall satisfies the differential equation m- mg-kv, where k is a constant that depends on the body's aerodynamic properties and the density of the air. (Assume dt that the fall is short enough so that the variation in the air's density will not affect the outcome...
The hyperbolic cosine and hyperbolic sine functions, f(x) cosh(x) and g(x) sinh(), are analogs of the trigonometric functions cos(x) and sin(z) and come up in many places in mathematics and its applications. (The hyperbolic cosine, for example, describes the curve of a hanging cable, called a catenary.) They are defined by the conditions cosh(0)-l, sinh(O), (cosh())inh("), d(sinh()- csh) (a) Using only this information, find the Taylor polynomial approximation for cosh(x) at0 of COS degree n = 4. (b) Using only...
Recall three facts about hyperbolic cosine and hyperbolic sine functions: dr cosh(x) = sinh(x), di sinh(x) = cosh(x), cosh? (x) – sinh? (x) = 1. Compute the Wronskian of cosh(2) and sinh(2). Using your question 1's answer, cosh() and sinh(x) are linearly independent on R True False
' ' 3. The hyperbolic trigonometric functions are defined by sinh(t) , The following identities might be helpful in the following problem. You should convince yourself that they are true, but you are not required to write up proofs/derivations cosh2 1-sinh2に1, cosh2 1 + sinh2にcosh(21), cosh 1 -sinh 1-cosh 1 sinh 1, dl dl (a) For a real number t, define sinh-1に1n(t + Vt2 + 1) Show that sinh(t) is the unique eaber u such that sinhu t (b) Use...
5. Use a substitution and an integration by parts to find each of the following indef- inite integrals: (b) | (cos(a) sin(a) esas) de (a) / ( (32 – 7) sin(5x + 2)) de (c) / (e* cos(e=)) dt (d) dr 6. Spot the error in the following calculation: S() will use integration by parts with 1 We wish to compute dr. For this dv du 1 dar = 1. This gives us dr by parts we find dr =...
question 10 and 11
10. Solve the equation 5 cosh x – 3 sinh x = 5. 11. (a) Show that cosh (x - y) = cosh x cosh y - sinh x sinh y. (b) Show that for any real number x, cosh” x + sinh? x = cosh2x . Hence prove that cosh2x = 2 sinhạ x + 1 = 2 cosh? x - 1. 1+ tanh x (c) Show that 1- tanh x
In class we discussed the relationship between the hyperbolic functions and a hyperbola then showed that it is analogous to that of the trigonometric functions and a circle a. Derive an analogue to the Pythagorean Identities (cos2 x + sin2 x 1, etc. ) for the hyperbolic functions hint: Which hyperbola and which circle? (this will give you the relationship between cosh x and sinh x and the others are then easily found as they were in the case of...
Problem 7. (20 points) We consider the function tanh(z) sinh(z) tanh(z)=cosh(z) where , For any integer 0, we denote by Qe the positively oriented square whose edges lie along the lines z-t(k+1) π and y = ± (k+)π -(km 4p. (a) Show that for any z z + iy e C, |cosh(z)12-sinh2(x) + cos2(y). 2p (b) Recall that tanh is analytic at the origin and that tanh () 1 - tanh2(). Compute the tanh(z) limit l := lim (Problem 7...
10. Find the first derivative with respect to the independent variable for the following functions: (a) f(x) = sinh(4x) (b) g(t) = cosh(t) sinh(t). 1 - cosh(r) (c) h(r) = 1 + cosh(r) (d) F(X) = tanh(e). (e) y = sinh--(VT). (Hint: Apply implicit differentiation to sinh(y) = VT.) 11. Use appropriate hyperbolic function substitutions to evaluate the following indef- inite integrals: dc (a) /+) Ke-10) (e)/() dx (b) dar James G., Modern Engineering Mathematics (5th ed.) 2015.: Exercise set...
If a body of mass m falling from rest under the action of gravity encounters an air resistance proportional to the square of the velocity, then the body's velocity t sec into dv the fall satisfies the differential equation m- mg-kv, where k is a constant that depends on the body's aerodynamic properties and the density of the air. (Assume dt that the fall is short enough so that the variation in the air's density will not affect the outcome...
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