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Calculus – Quick Reference Sheet

Limits and Continuity

Differential Calculus

Definitions
Limit: the value of the function as x approaches a certain value
Right-Hand Limit: the limit as x approaches from the positive direction; is noted as $x \rightarrow a^+$
Left-Hand Limit: the limit as x approaches from the negative direction; is noted as $x \rightarrow a^-$
Two-Sided Limit: the limit is the same whether x approaches from the right or left
Indeterminate Limits: Undefined limits such as 0 divided by 0, infinity divided by infinity, infinity times zero, infinity minus infinity, one to the power infinity, zero to the power zero and infinity to the power zero
Continuity: a function, f(x), is continuous over an interval if there exists a value of the function for every value of x on the interval
Discontinuity: a function, f(x), is discontinuous at a point if the function is not defined at that point
Types of Discontinuity
Removable Discontinuity: a limit of the function exists at the point of discontinuity
Jump Discontinuity: the right-handed and left-handed limits are different
Infinite Discontinuity: the limit of the function as x approaches the discontinuity is infinity
Finding Limits
By Substitution: limits for polynomial and rational functions with non-zero denominators can be found by substituting the value x into the function
By Factoring: limits for rational functions where the denominator is zero can be found by factoring the function and then simplifying
By Estimating: limits can be found for any function by using a graph or by calculating the value of f(x) for different values of that approach the limit
Limits and Asymptotes
Vertical Asymptote: a line, x=a that the graph of a function approaches but never touches $\lim_{x \to a}f(x)=\pm \infty$
Horizontal Asymptote: a line, y=a that the graph of a function approaches but never touches $\lim_{x \to \infty}f(x)=a$
Theorems
Intermediate Value Theorem: if the function is continuous between two points a and b, then there exists a value c between a and b where f(c) will be between f(a) and f(b)
Extreme Value Theorem: if a function is continuous over a closed interval then the function has a maximum and minimum on the interval
Sandwich or Squeezed Theorem: if a function f is between two other functions g and h with a common limit, then the function f has the same limit as g and h
Rules

Sum Rule $\[\lim_{x \to a}(f(x) \pm g(x)) = \lim_{x \to a} f(x) \pm \lim_{x \to a} g(x)$	Quotient Rule $\[\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a}f(x)}{\lim_{x \to a}g(x)}$
Product Rule $\[\lim_{x\to a}f(x)g(x) = \lim_{x \to a}f(x) \lim_{x \to a}g(x)$	Constant Multiple Rule $\[\lim_{x \to a}(k \cdot g(x)) = k \cdot \lim_{x \to a}g(x)$

Derivative: the function that describes the rate of change of another function
Geometrically, the derivative is the slope of a curve
Analytically, the derivative is the rate of change of a curve
Numerically, the derivative is the difference in y values on a very small interval of x
Higher Order Derivatives: taking the derivative of a derivative; the order of the derivative represents the number of times the derivative was taken
Partial Derivative: the derivative of only one variable in the function; other variables are treated as constants

Derivative Notation $\dfrac{df}{dx}$ or $f'(x)$

Second Order Derivative $\dfrac{d^2}{dx^2}f(x)$ or $f''(x)$

Theorems
Mean Value Theorem: if a function is continuous and differentiable on an interval [a,b], then there exists at least one point on that interval where the derivative is the slope through a and b.
Differentiability and Continuity
If a function is differentiable at a point then it is continuous at that point; the opposite is not always true
Differentiability and Inverse Functions If a function is differentiable at every point on an interval then it has an inverse function that is also differential on the interval
Finding Derivatives

Implicitly-Defined Functions:

Differentiate both sides of the equation with respect to x, treating y as a function of x.
Collect the terms with dy/dx on one side of the equation.
Factor out dy/dx.
Solve for dy/dx.

Inverse Functions

Write the inverse function relationship.
Differentiate both sides with respect to x.
Solve using implicit differentiation.

Inverse Functions:

Write the inverse function relationship
Differentiate both sides with respect to x.
Solve using implicit differentiation.

Rules

Sum Rule $\frac{d}{dx}(f(x) \pm g(x)) = f'(x) \pm g'(x)$	Constant Multiple Rule $\frac{d}{dx}(c(f(x)))=c \frac{d}{dx}f(x)$
Quotient Rule $\frac{d}{dx}(\frac{f(x)}{g(x)})=\frac{g(x)\cdot f'(x)-f(x)\cdot g'(x)}{(g(x))^2}$	Chain Rule $\frac{df}{dx}=\frac{df}{du} \cdot \frac{du}{dx}$
Product Rule $\frac{d}{dx}(f(x) \cdot g(x))=f'(x) \cdot g(x) + f(x) \cdot g'(x)$

Derivative Formulas

Polynomial Function
$\frac{d}{dx}(x^n)=nx^{n-1}$

Constant Function
$\frac{d}{dx}(c)=0$

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Integral Calculus

Application of Derivatives

Definitions
Integral: the process of finding the anti-derivative.
Anti-Derivative: the function F(x) such that the derivative is the original function f(x)
Indefinite Integral: the set of all anti-derivatives of f(x); the process of finding the anti-derivative of a function over no boundary
Definite Integral: the process of finding the anti-derivative of a function over a fixed boundary of x
Theorem
Fundamental Theorem of Calculus: a function that is continuous on a given interval will have an anti-derivative that is continuous on that interval and the integral of the derivative will be the function itself
Integral Approximation
Riemann Sum: the sum of all the rectangles that approximate the area under the curve
Midpoint Evaluation: the area between the curve and the x-axis is divided into equal rectangles; the function passes through the top midpoint of each rectangle
Left or Right Evaluation: the area between the curve and the x-axis is divided into equal rectangles; the function passesf through the top left or right corner of each rectangle
Trapezoidal Rule: the area between the curve and the x-axis is divided into trapezoids; the sum of the area of all trapezoids approximates the area under the curve.
Simpson’s Rule: the area between the curve and the x-axis is found by creating a parabola that intersects the curve at an even number of intervals of equal width;the area between the parabola and the x-axis will approximate the area of the curve.
Formulas

Indefinate Integrals $\int \! f(x)\, \mathrm{d}x = F(x)+C$	Definate Integrals $\int_a^b \! f(x)\, \mathrm{d}x = F(b)-F(a)$
Fund. Theorem of Calc. $F(x)=\frac{d}{dx} \! \int_a^x \!f(t) \ \mathrm{d}t =f(x)$	Area Under a Curve $\int_a^b \!f(x) \mathrm{d}x =\lim_{\|x\| \to 0}S_n$
Riemann Sum $\[\sum_{k=1}^n \! f(C_k)\cdot X_k$	Trapezoidal Rule $\int_a^b \! f(x) \mathrm{d}x \approx \frac{b-a}{2n}[f(x_0)+2f(x_1)+...+f(x_n)]$
Simpson’s Rule $\int_a^b \! f(x) \mathrm{d}x \approx \frac{b-a}{3n}[f(x_0)+4f(x_1)+2f(x_3)+...+f(x_n)]$

Properties
If f(x) and g(x) have an integral on [a,b] and k is a constant then the functions $k\cdot f(x)$ , $k\cdot g(x)$ and $f(x) \pm g(x)$ have an integral on [a,b].

Sum Rule $\int_a^b \![f(x)\pm g(x)] \mathrm{d}x=\int_a^b \! f(x) \mathrm{d}x \pm \int_a^b \! g(x) \mathrm{d}x$	Constant Multiple Rule $\int_a^b\! k \cdot f(x) \mathrm{d}x = k \int_a^b \! f(x) \mathrm{d}x$
Additive Interval Property If f(x) has an integral on three closed intervals [a,b], [b,c], and [a,c] then $\int_a^b \! f(x) \mathrm{d}x +\int_b^c \! f(x) \mathrm{d}x = \int_a^c \! f(x) \mathrm{d}x$	Preservation of Inequality If $f(x)\leq g(x)$ on [a,b] then $\int_a^b \! f(x) \mathrm{d}x \leq \int_a^b \! g(x) \mathrm{d}x$

Special Definate Integrals
If f(x) is defined at x=a then $\int_a^a \! f(x) \mathrm{d}x=0$
If f(x) has an integral on [a,b] then $\int_a^b \! f(x) \mathrm{d}x = - \int_b^a \! f(x) \mathrm{d}x$

Definitions
Local Maximum or Minimum: a point on the graph where derivative changes from positive to negative or negative to positive
Absolute Maximum: the highest point on the graph
Absolute Minimum: the lowest point on the graph
Monotonicity: a function that only increases or only decreases
Concavity: the rate of change of the slope of a curve
Velocity: the first derivative of a function describing position
Acceleration: the first derivative of a function describing velocity; the second derivative of function describing
position

Applications

Instantaneous Rate of Change
$f'(x_0)= \lim_{h \to 0} \frac{f(x_0+h)-f(x_0)}{h}$

Average Rate of Change
$\frac{\mathrm{d}y}{\mathrm{d}y}=\frac{f(x_0+h)-f(x_0)}{h}$

Slope of a Curve: the value of the derivative at a specific point
Optimization Problems: a problem that asks you to find a maximum or minimum of a quantity; write the equation for the quantity you are trying to optimize; find the derivative and set it equal to zero; solve

Relationship Between Graphs and Derivatives

	Negative	Zero	Positive
f(x)	Function is below x-axis	Function intersects the x-axis	Function is above x-axis
f'(x)	Function is decreasing	Function is flat	Function is increasing
f”(x)	Function is concave down	Point of inflection	Function is concave up

Application of Integration

Definitions
Average Value of a Function: the average of the value of a function over a given interval $av(f)=\frac{1}{b-a} \int_a^b \! f(x) \mathrm{d}x$
Differential Equation: an equation that relates a function to its anti-derivative
Separable Differential Equation: a differential equation where x and y variables can be separated
Volume: If f(x,y) has an integral over the plane region, R, and f(x,y)0 for all (x,y) in R then the volume, V, that is between the graph of f(x,y) and R is defined as
$V=\int_{Y_1}^{Y_2} \! \int_{X_1}^{X_2} \! f(x,y) \mathrm{d}x \! \mathrm{d}y=\int_{Y_1}^{Y_2} \! [F(X_2)-F(X_2)] \mathrm{d}y$
Relationships
Areas and Integrals: the integral over the interval [a,b] is the area under the curve of the function between the points a and b
Volumes and Integrals: the volume of a shape can be found by solving the double or triple integral of the function describing the volume.
Solving Differential Equations
Separable Differential Equations: combine like terms; evaluate the integral of both sides of the equation; solve for y.

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