Question

2. (a) Hyperbolic Sine and Hyperbolic Cosine are defined as $\sinh = \frac{e^x - e^{-x}}{2}$ and $\cosh = \frac{e^x + e^{-x}}{2}$. Show that $\cosh^2(x) - \sinh^2(x) = 1$. (b) What is the derivative of $\cosh(x)$? How about $\sinh(x)$? (c) Use the substitution $x = \sinh(y)$ to evaluate the integral $\int \frac{1}{\sqrt{1 + x^2}}$ (note that $\cosh(x) > 0$ for any $x$). Then, evaluate the integral using $x = \tan(\theta)$ and compare your answers to find a formula for a new function. When writing the formula, use the fact that $\sinh(0) = 0$ to eliminate any constants.

          2. (a) Hyperbolic Sine and Hyperbolic Cosine are defined as $\sinh = \frac{e^x - e^{-x}}{2}$ and $\cosh = \frac{e^x + e^{-x}}{2}$.
Show that $\cosh^2(x) - \sinh^2(x) = 1$.
(b) What is the derivative of $\cosh(x)$? How about $\sinh(x)$?
(c) Use the substitution $x = \sinh(y)$ to evaluate the integral $\int \frac{1}{\sqrt{1 + x^2}}$ (note that $\cosh(x) > 0$
for any $x$). Then, evaluate the integral using $x = \tan(\theta)$ and compare your answers to find
a formula for a new function. When writing the formula, use the fact that $\sinh(0) = 0$ to
eliminate any constants.

2. (a) Hyperbolic Sine and Hyperbolic Cosine are defined as sinh = (e^x - e^-x)/(2) and cosh = (e^x + e^-x)/(2).
Show that cosh^2(x) - sinh^2(x) = 1.
(b) What is the derivative of cosh(x)? How about sinh(x)?
(c) Use the substitution x = sinh(y) to evaluate the integral ∫(1)/(√(1 + x^2)) (note that cosh(x) > 0
for any x). Then, evaluate the integral using x = tan(θ) and compare your answers to find
a formula for a new function. When writing the formula, use the fact that sinh(0) = 0 to
eliminate any constants.

Added by Ashley K.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Israel Hernandez

04/23/2022

Step 1

Step 1: To show that cosh^2(x) - sinh^2(x) = 1, we can start by using the definitions of cosh and sinh: cosh(x) = (e^x + e^(-x))/2 sinh(x) = (e^x - e^(-x))/2 Show more…

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(a) Hyperbolic Sine and Hyperbolic Cosine are defined as sinh=(e^(x)-e^(-x))/(2) and cosh=(e^(x)+e^(-x))/(2). Show that cosh^(2)(x)-sinh^(2)(x)=1. (b) What is the derivative of cosh(x) ? How about sinh(x) ? (c) Use the substitution x=sinh(y) to evaluate the integral int (1)/(sqrt(1+x^(2))) (note that cosh(x)>0 for any x ). Then, evaluate the integral using x=tan( heta ) and compare your answers to find a formula for a new function. When writing the formula, use the fact that sinh(0)=0 to eliminate any constants. (a) Hyperbolic Sine and Hyperbolic Cosine are defined as sinh = e* Show that cosh2-sinh2=1. and cosh= e"+e 2 (b) What is the derivative of cosh()? How about sinh()? c Use the substitution x = sinh(y to evaluate the integral note that cosh(> 0 for any x). Then, evaluate the integral using x = tan(0) and compare your answers to find a formula for a new function. When writing the formula, use the fact that sinh(0) = 0 to eliminate any constants.