Notes

Practice Problems

\(\textbf{1)}\) Simplify \(\cosh{x}+\sinh{x}\)

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The answer is \( e^x \)

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\(\,\,\,\,\,\,\cosh{x}+\sinh{x}\)

\(\,\,\,\,\,\,\displaystyle \frac{1}{2}\left(e^x+e^{-x}\right)+\frac{1}{2}\left(e^x-e^{-x}\right)\)

\(\,\,\,\,\,\,\displaystyle \frac{1}{2}e^x+\frac{1}{2}e^{-x}+\frac{1}{2}e^x-\frac{1}{2}e^{-x}\)

\(\,\,\,\,\,\,\displaystyle \frac{1}{2}e^x+\frac{1}{2}e^x\)

The answer is \( e^x \)

\(\textbf{2)}\) Solve for x, \(\sinh{x}=\frac{5}{12}\)

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The answer is \( x=\ln{\left(\frac{3}{2}\right)} \)

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\(\,\,\,\,\,\,\sinh{x}=\displaystyle \frac{5}{12}\)

\(\,\,\,\,\,\,\displaystyle \frac{1}{2}\left(e^x-e^{-x}\right)=\frac{5}{12}\)

\(\,\,\,\,\,\,\displaystyle \frac{1}{2}\left(e^x-e^{-x}\right)-\frac{5}{12}=0\)

\(\,\,\,\,\,\,6e^x-6e^{-x}-5=0\)

\(\,\,\,\,\,\,6e^2x-6-5e^x=0\)

\(\,\,\,\,\,\,6e^2x-5e^x-6=0\)

\(\,\,\,\,\,\,\left(2e^x-3\right)\left(3e^x+2\right)=0\)

\(\,\,\,\,\,\,\left(2e^x-3\right)=0 \text{ or }\left(3e^x+2\right)=0\)

\(\,\,\,\,\,\,2e^x-3=0 \text{ or }3e^x+2=0\)

\(\,\,\,\,\,\,2e^x=3 \text{ or }3e^x=-2\)

\(\,\,\,\,\,\,e^x=\displaystyle \frac{3}{2} \text{ or }e^x=-\frac{2}{3}\)

\(\,\,\,\,\,\,\displaystyle x=\ln{\frac{3}{2}} \text{ or Not Possible}\)

The answer is \( \displaystyle x=\ln{\frac{3}{2}} \)

\(\textbf{3)}\) Solve for x, \(\cosh{(x)}+5\sinh{(x)}=5\)

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The answer is \( x=ln(2) \)

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\(\,\,\,\,\,\,\cosh{(x)}+5\sinh{(x)}=5\)

\(\,\,\,\,\,\,\displaystyle \frac{1}{2}\left(e^x+e^{-x}\right)+5\cdot\frac{1}{2}\left(e^x-e^{-x}\right)=5\)

The answer is \( x=ln(2) \)

\(\textbf{4)}\) Solve for x, \(5\sinh\left(4x\right)-2\cosh\left(4x\right)=10\)

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The answer is \( x= 1/4 \ln{(7)} \)

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\(5\sinh\left(4x\right)-2\cosh\left(4x\right)=10\)

The answer is \( x= 1/4 \ln{(7)} \)

\(\textbf{5)}\) Find dy/dx of \(y=6\cosh{(x)}\)

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The answer is \( dy/dx= 6 \sinh{(x)} \)

\(\textbf{6)}\) Evaluate \(\sinh 0\)

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The answer is \(\sinh 0 = 0\)

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\(\,\,\,\,\,\,\sinh x=\displaystyle \frac{e^x – e^{-x}}{2}\)

\(\,\,\,\,\,\,\sinh 0=\displaystyle \frac{e^0 – e^{-0}}{2}\)

\(\,\,\,\,\,\,\sinh 0=0\)

\(\textbf{7)}\) Evaluate \(\cosh 0\)

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The answer is \(\cosh 0 = \displaystyle \frac{e^0 + e^0}{2} = 1\)

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\(\,\,\,\,\,\,\cosh x=\displaystyle \frac{e^x + e^{-x}}{2}\)

\(\,\,\,\,\,\,\cosh 0=\displaystyle \frac{e^0 + e^0}{2}\)

\(\,\,\,\,\,\,\cosh 0=\displaystyle \frac{1 + 1}{2}\)

\(\,\,\,\,\,\,\cosh 0=1\)

\(\textbf{8)}\) Evaluate \(\tanh 0\)

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The answer is \(\tanh 0 = \displaystyle \frac{\sinh 0}{\cosh 0} = \frac{0}{1} = 0\)

\(\textbf{9)}\) Evaluate \(\sinh(\ln 2)\)

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The answer is \(\sinh(\ln 2) = \displaystyle \frac{3}{4} \)

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\(\,\,\,\,\,\,\sinh (x)=\displaystyle \frac{e^x – e^{-x}}{2}\)

\(\,\,\,\,\,\,\sinh(\ln 2) = \displaystyle \frac{e^{\ln 2} – e^{-\ln 2}}{2} \)

\(\,\,\,\,\,\,\sinh(\ln 2) = \displaystyle \frac{2 – \frac{1}{2}}{2} \)

\(\,\,\,\,\,\,\sinh(\ln 2) = \displaystyle \frac{\frac{3}{2}}{2} \)

\(\,\,\,\,\,\,\sinh(\ln 2) = \displaystyle \frac{3}{4} \)

\(\textbf{10)}\) Evaluate \(\cosh(\ln 3)\)

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The answer is \(\cosh(\ln 3) =\frac{5}{3}\)

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\(\,\,\,\,\,\,\cosh (x)=\displaystyle \frac{e^x + e^{-x}}{2}\)

\(\,\,\,\,\,\,\cosh (\ln 3)=\displaystyle \frac{e^{\ln 3} + e^{-\ln 3}}{2}\)

\(\,\,\,\,\,\,\cosh (\ln 3)=\displaystyle \frac{3 + \frac{1}{3}}{2}\)

\(\,\,\,\,\,\,\cosh (\ln 3)=\displaystyle \frac{\frac{10}{3}}{2}\)

\(\,\,\,\,\,\,\cosh (\ln 3)=\displaystyle \frac{5}{3}\)

\(\textbf{11)}\) Is the function \(\sinh x \) even, odd or neither?

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The answer is \(\sinh(x)\) is odd.

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\(\,\,\,\,\,\,f(-x)=-f(x)\) (definition of odd function)

\(\,\,\,\,\,\,\sinh(-x)=-\sinh(x)\)

\(\,\,\,\,\,\,\displaystyle \frac{e^{-x} – e^x}{2}=-\frac{e^x – e^{-x}}{2}\)

\(\,\,\,\,\,\,\displaystyle \frac{e^{-x} – e^x}{2}=\frac{-e^x + e^{-x}}{2}\)

\(\,\,\,\,\,\,\displaystyle \frac{e^{-x} – e^x}{2}=\frac{e^{-x} – e^x}{2}\)

\(\,\,\,\,\,\, \sinh x\) is odd.

\(\textbf{12)}\) Is the function \(\cosh x \) even, odd or neither?

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The answer is even \(\cosh(-x) = \displaystyle \frac{e^{-x} + e^x}{2} = \frac{e^x + e^{-x}}{2} = \cosh x\)

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\(\,\,\,\,\,\,f(x)=f(-x)\) (definition of even function)

\(\,\,\,\,\,\,\cosh(x)=\cosh(-x)\)

\(\,\,\,\,\,\,\displaystyle \frac{e^{x} + e^{-x}}{2}= \frac{e^{-x} + e^{-(-x)}}{2}\)

\(\,\,\,\,\,\,\displaystyle \frac{e^{x} + e^{-x}}{2}= \frac{e^{-x} + e^{x}}{2}\)

\(\,\,\,\,\,\,\displaystyle \frac{e^{x} + e^{-x}}{2}= \frac{e^{x} + e^{-x}}{2}\)

\(\,\,\,\,\,\, \cosh x\) is even.

See Related Pages\(\)

\(\bullet\text{ Calculus Homepage}\)
\(\,\,\,\,\,\,\,\,\text{All the Best Topics…}\)

\(\bullet\text{ Definition of Derivative}\)
\(\,\,\,\,\,\,\,\, \displaystyle \lim_{\Delta x\to 0} \frac{f(x+ \Delta x)-f(x)}{\Delta x} \)

\(\bullet\text{ Equation of the Tangent Line}\)
\(\,\,\,\,\,\,\,\,f(x)=x^3+3x^2−x \text{ at the point } (2,18)\)

\(\bullet\text{ Derivatives- Constant Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}(c)=0\)

\(\bullet\text{ Derivatives- Power Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}(x^n)=nx^{n-1}\)

\(\bullet\text{ Derivatives- Constant Multiple Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}(cf(x))=cf'(x)\)

\(\bullet\text{ Derivatives- Sum and Difference Rules}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}[f(x) \pm g(x)]=f'(x) \pm g'(x)\)

\(\bullet\text{ Derivatives- Sin and Cos}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}sin(x)=cos(x)\)

\(\bullet\text{ Derivatives- Product Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}[f(x) \cdot g(x)]=f(x) \cdot g'(x)+f'(x) \cdot g(x)\)

\(\bullet\text{ Derivatives- Quotient Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}\left[\displaystyle\frac{f(x)}{g(x)}\right]=\displaystyle\frac{g(x) \cdot f'(x)-f(x) \cdot g'(x)}{[g(x)]^2}\)

\(\bullet\text{ Derivatives- Chain Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}[f(g(x))]= f'(g(x)) \cdot g'(x)\)

\(\bullet\text{ Derivatives- ln(x)}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}[ln(x)]= \displaystyle \frac{1}{x}\)

\(\bullet\text{ Implicit Differentiation}\)
\(\,\,\,\,\,\,\,\,\)

\(\bullet\text{ Horizontal Tangent Line}\)
\(\,\,\,\,\,\,\,\,\)

\(\bullet\text{ Mean Value Theorem}\)
\(\,\,\,\,\,\,\,\,\)

\(\bullet\text{ Related Rates}\)
\(\,\,\,\,\,\,\,\,\)

\(\bullet\text{ Increasing and Decreasing Intervals}\)
\(\,\,\,\,\,\,\,\,\)

\(\bullet\text{ Intervals of concave up and down}\)
\(\,\,\,\,\,\,\,\,\)

\(\bullet\text{ Inflection Points}\)
\(\,\,\,\,\,\,\,\,\)

\(\bullet\text{ Graph of f(x), f'(x) and f”(x)}\)
\(\,\,\,\,\,\,\,\,\)

\(\bullet\text{ Newton’s Method}\)
\(\,\,\,\,\,\,\,\,x_{n+1}=x_n – \displaystyle \frac{f(x_n)}{f'(x_n)}\)

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