# Part 1: Functions Defined by Derivatives and Integrals

## 1.1 Logarithms and Exponential Functions

#### Textbook References

• University Calculus: Early Transcendentals (3rd Ed)
• 7.1 (review: 1.5,1.6)

### 1.1.1 The Natural Logarithm

• Using integrals, we may rigorously define a logarithm.
• $$\ln x=\int_1^x \frac{1}{t}\,dt$$ for all $$x>0$$
• $$\frac{d}{dx}[\ln x] = \frac{1}{x}$$ and $$\ln 1 = 0$$
• 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.)
• Example. Find $$\int \frac{3}{x^2}-\frac{2}{x}+1+4x^2\,dx$$.

### 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

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.)
2. Find $$\int \frac{6}{x^3}+\frac{2}{x}-3x\,dx$$.
3. Find $$\int \frac{6x^4-x^2+4}{2x^3}\,dx$$.
4. We saw that $$\frac{d}{dx}[e^x]=e^x$$. Describe infinitely many other functions $$f(x)$$ such that $$f’(x)=f(x)$$.
5. Find $$\frac{d}{dx}[\frac{1}{x}+3e^x]$$.
6. Prove the following derivative formulas: $$\frac{d}{dx}[\log_b x]=\frac{1}{x\ln b}$$ and $$\frac{d}{dx}[a^x]=a^x \ln a$$.
7. (Quiz) Integrate $$\int 3x^4+3e^x-\frac{4}{x}\,dx$$.
• $$12x^3-3e^x+4\ln|x|+C$$
• $$\frac{3}{5}x^5+3e^x-4\ln|x|+C$$
• $$\frac{3}{4}x^5+3xe^{x-1}-\frac{4}{x^2}+C$$
• None of the above
8. (Quiz) Differentiate $$f(x)=\ln(x^2)+e^{x^3}$$.
• $$f’(x)=\frac{2}{x}+3x^2e^{x^3}$$
• $$f’(x)=2x\ln(x^2)+x^3e^{x^3-1}+C$$
• $$f’(x)=\frac{2x}{x^2}+e^{x^3}$$
• None of the above

Solutions

## 1.2 Trigonometric Functions

#### Textbook References

• 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

1. Show that $$f(x)=\cos(x)$$ is a solution to the differential equation $$f^{\prime\prime}(x)=-f(x),f’(0)=0,f(0)=1$$.
2. Show that $$f(x)=\sin(3x)$$ is a solution to the differential equation $$f^{\prime\prime}(x)=-9f(x),f’(0)=3,f(0)=0$$.
3. Find a solution to the differential equation $$f^{\prime\prime}(x)=-f(x),f’(0)=0,f(0)=4$$.
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.)
5. (Optional) Find a solution to the differential equation $$f^{\prime\prime}(x)=-4f(x),f’(0)=6,f(0)=0$$.
6. (Optional) Prove that $$-1\leq\sin x\leq 1$$ and $$-1\leq\cos x\leq 1$$. (Hint: use the Pythagorean identity.)

Solutions

## 1.3 Hyperbolic Functions

#### Textbook References

• University Calculus: Early Transcendentals (3rd Ed)
• 7.3

### 1.3.1 Hyperbolic Sine and Cosine

• Exponential functions are used to define the hyperbolic functions, which behave like trigonometric functions in many ways.
• $$\sinh x = \frac{e^x-e^{-x}}{2}$$
• $$\cosh x = \frac{e^x+e^{-x}}{2}$$
• Example Evaluate $$\sinh(0)$$ and $$\cosh(0)$$.
• Example Evaluate $$\cosh(\ln 4)$$.
• Example Prove that $$\sinh(2x)=2\sinh(x)\cosh(x)$$.


### 1.3.2 Other Hyperbolic Functions

• 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

1. Evaluate $$\sinh(\ln 6)$$.
2. Prove that $$\cosh (2x) = \cosh^2 x + \sinh^2 x$$.
3. Prove that $$\cosh^2 x - \sinh^2 x = 1$$.
4. Evaluate $$\tanh(\ln 3)$$.
5. Simplify $$\sinh(x)\coth(x)\cosh(x)-\frac{1}{\csch^2(x)}$$. (Hint: convert everything to $$\sinh x$$ and $$\cosh x$$.)
6. Prove that $$\frac{d}{dx}[\sinh x] = \cosh x$$.
7. Prove that $$\frac{d}{dx}[\sech x] = -\sech x\tanh x$$. (Hint: use the fact that $$\frac{d}{dx}[\cosh x] = \sinh x$$.)
8. Compute $$\frac{d}{dx}[\tanh(3x)-\sech(\ln x)]$$.
9. Find $$\int 3\csch x\coth x - 2\sinh x\,dx$$.
10. (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 $$\frac{d}{dx}[\sinh^{\leftarrow}(x)]=\frac{1}{\sqrt{1+x^2}}$$.
11. (Optional) Prove that $$\sinh^{\leftarrow}(x)=\ln(\sqrt{x^2+1}+x)$$.
12. (Quiz) Evaluate $$\cosh(\ln 2)$$.
• $$\frac{3}{5}$$
• $$\frac{2}{3}$$
• $$\frac{5}{4}$$
13. (Quiz) Differentiate $$f(x)=\tanh(x^2)-\cosh(2x+1)$$.
• $$f’(x)=-\sech(x^2)\tanh(x^2)+\sinh(2x+1)+2$$
• $$f’(x)=2x\sech^2(x^2)-2\sinh(2x+1)$$
• $$f’(x)=\frac{1}{1+x^4}+2\sinh(2x+1)$$

## Review Exercises

The exercises are now located with their respective notes.