Example.
Use the definition \( \ln x=\int_1^x \frac{1}{t}\,dt \)
to prove the property \( \ln(ax) = \ln a + \ln x \) for
\(a,x>0\). (Hint: start by showing that the derivatives are the same.)
This allows us to express an indefinite integral for \(1/x\):
\(\int\frac{1}{x}\,dx=\ln|x|+C\). (Note the absolute value.)
1.1.2 The Natural Number and Natural Exponential Function
Note that \(a^p\) has only been defined for when \(p\in\mathbb Q\).
Since \(f(x)=\ln x\) is differentiable and 1-to-1, we can define
\(\exp x=f^{\leftarrow}(x)\) to be its differentiable inverse.
Example.
Use the fact
\(\frac{d}{dx}[f^{\leftarrow}(x)]=\frac{1}{f’(f^{\leftarrow}(x))}\)
to prove that \(\frac{d}{dx}[\exp x]=\exp x\).
(Hint: let \(f(x)=\ln x,f’(x)=\frac{1}{x},f^{\leftarrow}(x)=\exp x\).)
Let \(e=\exp 1\). We’ll see much later in the
course why \(e\approx 2.718\).
1.1.3 General Logarithms and Exponential Functions
Since \(\exp x\) is defined for all real numbers, we may define
\(a^x = \exp(x\ln a)\) for all \(a>0\) and \(x\in\mathbb R\).
Note that \(e^x = \exp x\).
Example.
Use the definition \(a^x = \exp(x\ln a)\) and
property \(\ln(abc)=\ln a + \ln b + \ln c\) to show that
\(2^3 = 2\times2\times2\).
Define \( \log_b x = \frac{\ln x}{\ln b} \) for \(b>1\).
Example.
Use the definitions
\( \log_b x = \frac{\ln x}{\ln b} \) and \(b^x = \exp(x\ln b)\)
to prove the property \( x = \log_b(b^x) \). (That is,
\(\log_b x\) and \(b^x\) are inverse functions.)
Exercises for 1.1
Use the definition \( \ln x=\int_1^x \frac{1}{t}\,dt \)
to prove the property \( \ln(x^p) = p\ln x \) for \(x>0\)
and \(p\in\mathbb Q\). (Hint: start by showing that both sides share
the same derivative.)
Find \(\int \frac{6}{x^3}+\frac{2}{x}-3x\,dx\).
Find \(\int \frac{6x^4-x^2+4}{2x^3}\,dx\).
We saw that \(\frac{d}{dx}[e^x]=e^x\).
Describe infinitely many other functions \(f(x)\) such that
\(f’(x)=f(x)\).
Find \(\frac{d}{dx}[\frac{1}{x}+3e^x]\).
Prove the following derivative formulas:
\( \frac{d}{dx}[\log_b x]=\frac{1}{x\ln b} \) and
\( \frac{d}{dx}[a^x]=a^x \ln a \).
University Calculus: Early Transcendentals (3rd Ed)
none (review: 1.3)
1.2.1 Sine and Cosine
Similar to how \(\ln x\) and \(e^x\) are defined by integrals, we use
derivatives to define the trigonometric functions.
Define \(f(x)=\sin x\) to be the unique solution to the differential equation
(initial value problem) \(f^{\prime\prime}(x)=-f(x),f’(0)=1,f(0)=0\).
Define \(\cos(x)\) to be the derivative \(\frac{d}{dx}[\sin x]\).
Example
Prove that \(\frac{d}{dx}[\cos x]=-\sin x\).
The other four trig functions are then defined as usual as quotients
of \(\sin x\) and \(\cos x\).
1.2.2 Geometric Properties
The geometric properties of \(\sin x\) and \(\cos x\) come from the
fact that they satisfy the Pythagorean identity:
\([\sin x]^2+[\cos x]^2=1\).
Example
Prove the Pythagorean identity. (Hint: start by showing that both sides share
the same derivative.)
Exercises for 1.2
Show that \(f(x)=\cos(x)\) is a solution to the differential equation
\(f^{\prime\prime}(x)=-f(x),f’(0)=0,f(0)=1\).
Show that \(f(x)=\sin(3x)\) is a solution to the differential equation
\(f^{\prime\prime}(x)=-9f(x),f’(0)=3,f(0)=0\).
Find a solution to the differential equation
\(f^{\prime\prime}(x)=-f(x),f’(0)=0,f(0)=4\).
Prove that if \( x\) is an angle where
\(\sin x = -\frac{5}{13}\), then \(\cos x\) is either
\(\frac{12}{13}\) or \(-\frac{12}{13}\). (Hint: use the Pythagorean
identity.)
(Optional)
Find a solution to the differential equation
\(f^{\prime\prime}(x)=-4f(x),f’(0)=6,f(0)=0\).
(Optional)
Prove that \(-1\leq\sin x\leq 1\) and \(-1\leq\cos x\leq 1\).
(Hint: use the Pythagorean identity.)
The other hypberbolic functions are defined the same way their
trig counterparts are, and have similar properties.
\(\tanh x = \frac{\sinh x}{\cosh x} = \frac{e^x-e^{-x}}{e^x+e^{-x}}\)
\(\coth x = \frac{\cosh x}{\sinh x} = \frac{e^x+e^{-x}}{e^x-e^{-x}}\)
\(\sech x = \frac{1}{\cosh x}=\frac{2}{e^x+e^{-x}}\)
\(\csch x = \frac{1}{\sinh x}=\frac{2}{e^x-e^{-x}}\)
Example Evaluate \(\sech(-\ln 2)\).
Example Prove that \(\tanh^2(x)=1-\sech^2(x)\).
1.3.3 Derivatives and Integrals of Hyperbolic Functions
Their derivatives also behave similarly.
\(\frac{d}{dx}[\sinh x] = \cosh x\)
\(\frac{d}{dx}[\cosh x] = \sinh x\)
\(\frac{d}{dx}[\tanh x] = \sech^2 x\)
\(\frac{d}{dx}[\coth x] = -\csch^2 x\)
\(\frac{d}{dx}[\sech x] = -\sech x\tanh x\)
\(\frac{d}{dx}[\csch x] = -\csch x\coth x\)
Example
Use their definitions to prove that \(\frac{d}{dx}[\cosh x]=\sinh x\).
Example
Use their definitions to prove that \(\frac{d}{dx}[\coth x]=\csch^2 x\).
Example
Compute \(\frac{d}{dx}[\sinh(2x)+\coth(x^2)]\).
Their integral formulas may be found by just reversing the equations.
\(\int \cosh x\,dx = \sinh x + C \)
\(\int \sinh x\,dx = \cosh x + C \)
\(\int \sech^2 x\,dx = \tanh x + C \)
\(\int \csch^2 x\,dx = -\coth x + C \)
\(\int\sech x\tanh x\,dx = -\sech x + C \)
\(\int\csch x\coth x\,dx = -\csch x + C \)
Example
Find \(\int 4\csch^2 x-3\sinh x\,dx\).
Exercises for 1.3
Evaluate \(\sinh(\ln 6)\).
Prove that \(\cosh (2x) = \cosh^2 x + \sinh^2 x\).
Prove that \(\cosh^2 x - \sinh^2 x = 1\).
Evaluate \(\tanh(\ln 3)\).
Simplify \(\sinh(x)\coth(x)\cosh(x)-\frac{1}{\csch^2(x)}\).
(Hint: convert everything to \(\sinh x\) and \(\cosh x\).)
Prove that \(\frac{d}{dx}[\sinh x] = \cosh x\).
Prove that \(\frac{d}{dx}[\sech x] = -\sech x\tanh x\).
(Hint: use the fact that \(\frac{d}{dx}[\cosh x] = \sinh x\).)
Compute \(\frac{d}{dx}[\tanh(3x)-\sech(\ln x)]\).
Find \(\int 3\csch x\coth x - 2\sinh x\,dx\).
(Optional)
Let \(\sinh^{\leftarrow}(x)\) be the inverse function of
\(\sinh(x)\). Use the facts
\(\frac{d}{dx}[f^{\leftarrow}(x)]=\frac{1}{f’(f^{\leftarrow}(x))}\)
and \(\cosh^2 x-\sinh^2 x = 1\) to prove that
\(\frac{d}{dx}[\sinh^{\leftarrow}(x)]=\frac{1}{\sqrt{1+x^2}}\).
(Optional)
Prove that \(\sinh^{\leftarrow}(x)=\ln(\sqrt{x^2+1}+x)\).