Lecture #1: Calculus I Review
MATH 146 – Calculus II
January 5, 2026
Limits
Given a function \(f\), a fixed input \(x=a\), and a real number \(L\), we say that \(f\) has limit \(L\) as \(x\) approaches \(a\), and write
\[ \lim_{x \to a} f(x) = L, \]
provided that we can make \(f(x)\) as close to \(L\) as we like by taking \(x\) sufficiently close (but not equal) to \(a\).
If we cannot make \(f(x)\) as close to a single value as we would like as \(x\) approaches \(a\), then we say that \(f\) does not have a limit as \(x\) approaches \(a\).
Limits Example
Left-Hand Limits
We say that \(f\) has limit \(L_1\) as \(x\) approached \(a\) from the left and write
\[ \lim_{x \to a^-} f(x) = L_1, \]
provided that we can make the value of \(f(x)\) as close to \(L_1\) as we like by taking \(x\) sufficiently close to \(a\), while always having \(x < a\).
We call \(L_1\) the left-hand limit of \(f\) as \(x\) approaches \(a\).
Right-Hand Limits
We say that \(f\) has limit \(L_2\) as \(x\) approached \(a\) from the right and write
\[ \lim_{x \to a^+} f(x) = L_2, \]
provided that we can make the value of \(f(x)\) as close to \(L_2\) as we like by taking \(x\) sufficiently close to \(a\), while always having \(x > a\).
We call \(L_1\) the right-hand limit of \(f\) as \(x\) approaches \(a\).
Left-Hand and Right-Hand Limits
A function \(f\) has limit \(L\) as \(x \to a\) if and only if
\[ \lim_{x \to a^-} f(x) = L = \lim_{x \to a^+} f(x). \]
Limits Example
Continuity
What does it mean for a function to be continuous? A function \(f\) is continuous at \(x=a\) provided that:
- \(f\) has a limit as \(x \to a\),
- \(f\) is defined at \(x=a\), and
- \(\lim\limits_{x \to a} f(x) = f(a)\).
Limits Example
The Limit Does Not Exist!
The Derivative Function
This leads us to the following definition:
Let \(f\) be a function and \(x\) a value in the function’s domain.
We define the derivative of \(f\), a new function called \(f'\), by the formula
\[ f'(x) = \lim\limits_{h \to 0} \frac{f(x + h) - f(x)}{h}, \]
provided this limit exists.
\(f'(x)\) is sometimes written as \(\dfrac{d}{dx} f(x)\).
Application of the Derivative
We say that \(f\) is increasing on \((a, b)\) provided that the function is always rising as we move from left to right. That is, for any \(x\) and \(y\) in \((a,b)\), if \(x < y\), then \(f(x) < f(y)\).
We say that \(f\) is decreasing on \((a, b)\) provided that the function is always falling as we move from left to right. That is, for any \(x\) and \(y\) in \((a,b)\), if \(x < y\), then \(f(x) > f(y)\).
Application of the Derivative
Let \(f\) be a function that is differentiable on an interval \((a, b)\). If \(f'(x) > 0\) for every \(x\) such that \(a < x < b\), then \(f\) is increasing on \((a, b)\).
Let \(f\) be a function that is differentiable on an interval \((a, b)\). If \(f'(x) < 0\) for every \(x\) such that \(a < x < b\), then \(f\) is decreasing on \((a, b)\).
Differentiability
We say that a function \(f\) is differentiable at \(x = a\) if it has a derivative at \(x = a\).
If \(f\) is differentiable at \(x = a\), the \(f\) is continuous at \(x = a\). Equivalently, if \(f\) fails to be continuous at \(x = a\), the \(f\) will not be differentiable at \(x = a\).
A function can be continuous at a point, but not be differentiable there. In particular, a function \(f\) is not differentiable at \(x = a\) if the graph has a sharp corner (or cusp) at the point \((a, f(a))\).
If \(f\) is differentiable at \(x = a\), the \(f\) is locally linear at \(x = a\). That is, when a function is differentiable, it looks linear when viewed up close because it resembles its tangent line there.
Example: Continuity and Differentiability
The Constant Multiple Rule
For any real number \(k\), if \(f(x)\) is a differentiable function with derivative \(f'(x)\), then \[ \frac{d}{dx} \left[ k f(x) \right] = k f'(x). \]
The Sum Rule
If \(f(x)\) and \(g(x)\) are differentiable functions with derivatives \(f'(x)\) and \(g'(x)\) respectively, then \[ \frac{d}{dx} [f(x) + g(x)] = f'(x) + g'(x). \]
The Power Rule
For any nonzero real number \(n\), if \(f(x) = x^n\), then \[ f'(x) = nx^{n-1}. \]
The Product Rule
If \(f\) and \(g\) are differentiable functions, then their product \(P(x) = (f \cdot g)(x) = f(x) \cdot g(x)\) is also a differentiable function, and
\[ P'(x) = f(x) g'(x) + g(x)f'(x). \]
The Quotient Rule
If \(f\) and \(g\) are differentiable functions, then their quotient \(Q(x) = \frac{f(x)}{g(x)}\) is also a differentiable function for all \(x\) where \(g(x) \ne 0\), where
\[ Q'(x) = \frac{g(x)f'(x) - f(x)g'(x)}{\left(g(x)\right)^2}. \]
The Chain Rule
If \(f\) and \(g\) are differentiable functions, then:
\[ \frac{d}{dx} (f \circ g)(x) = \frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x). \]
Derivatives of Trigonometric Functions
- \(\dfrac{d}{dx} \sin(x) = \cos(x)\).
- \(\dfrac{d}{dx} \cos(x) = -\sin(x)\).
- \(\dfrac{d}{dx} \tan(x) = \sec^2(x)\).
- \(\dfrac{d}{dx} \sec(x) = \sec(x) \tan(x)\).
- \(\dfrac{d}{dx} \csc(x) = -\csc(x) \cot(x)\).
- \(\dfrac{d}{dx} \cot(x) = -\csc^2(x)\).
Derivatives of Inverse Trigonometric Functions
- \(\dfrac{d}{dx} \sin^{-1}(x) = \dfrac{1}{\sqrt{1 - x^2}}, \, \text{for } \lvert x \rvert < 1\).
- \(\dfrac{d}{dx} \cos^{-1}(x) = -\dfrac{1}{\sqrt{1 - x^2}}, \, \text{for } \lvert x \rvert < 1\).
- \(\dfrac{d}{dx} \tan^{-1}(x) = \dfrac{1}{1+x^2}\).
- \(\dfrac{d}{dx} \sec^{-1}(x) = \dfrac{1}{\lvert x \rvert \sqrt{x^2 - 1}}, \, \text{for } \lvert x \rvert > 1\).
- \(\dfrac{d}{dx} \csc^{-1}(x) = -\dfrac{1}{\lvert x \rvert \sqrt{x^2 - 1}}, \, \text{for } \lvert x \rvert > 1\).
- \(\dfrac{d}{dx} \cot^{-1}(x) = - \dfrac{1}{1+x^2}\).
Derivatives of Exponential & Logarithmic Functions
- \(\dfrac{d}{dx} e^x = e^x\).
- \(\dfrac{d}{dx} a^x = a^x \ln\left(a\right)\).
- \(\dfrac{d}{dx} \ln\left(x\right) = \dfrac{1}{x}, \, x > 0\).
- \(\dfrac{d}{dx} \log_{a}\left(x\right) = \dfrac{1}{x \ln\left(a\right)}, \, x > 0\).
Antiderivatives and Integral Notation
If \(f(x)\) is a function, then an antiderivative of \(f(x)\) is a function \(F(x)\) such that \(F'(x) = f(x)\).
If \(f(x)\) is a function, then the most general antiderivative of \(f(x)\) is denoted by: \[ \int f(x)\,dx. \]
Example: \[ \int 2x\,dx = x^2 + C. \]
Rules for Antiderivatives
Rules for Finding Antiderivatives:
- (Inverse) Power Rule:
\[ \int x^n\,dx = \frac{x^{n+1}}{n+1} + C,\textrm{for all $n \ne -1$}. \]
- Sum Rule:
\[ \int \left( f(x) + g(x) \right)\,dx = \int f(x)\,dx + \int g(x)\,dx. \]
Rules for Antiderivatives
Rules for Finding Antiderivatives:
- Constant Multiple Rule:
\[ \int cf(x)\,dx = c \int f(x)\,dx. \]
Differential Equations
A differential equation is an equation that relates an unknown function \(y(x)\) and one or more of its derivatives.
A differential equation with an initial condition \(y(x_0) = y_0\) is an initial value problem.
