Calculus

Module Co-ordinator: mgreenall@lincoln.ac.uk (Martin Greenall) Description of module

Read Calculus by J. Stewart (for the syllabus), Calculus by M. Spivak (for the background), Engineering Mathematics by K. A. Stroud (for the worked examples)

Course Components

Coursework Assignment (7%)
WebAssign Assignment x4 (8%)
In-Class Assignment (25%)
Final Exams (60%)

Outline Syllabus

Learning Outcomes

LO1 Formulate the basic concepts of differential and integral calculus.
LO2 Apply properties of derivative and its geometric and mechanical interpretations
for sketching graphs of functions.
LO3 State the fundamental theorem of calculus on the connection between derivative and integral; apply this theorem in specific examples.
LO4 Formulate the definitions of converging series and use Taylor series for analysing functions.

Flashcards

Automatically ported into Anki TARGET DECK University::MTH1002 Calculus

STARTI [Basic] What is the MTH1002 module? Back: Calculus. ENDI
STARTI [Basic] Question: What is a function? Back: A function is a mathematical rule that transforms an element from one set into an element of another set. Specifically, a function (f) from a set (X) to a set (Y) assigns a unique element (y \in Y) to each element (x \in X). ENDI
STARTI [Basic] Question: What is the domain of a function? Back: The domain (X) of a function is the set of all possible input values, i.e., the values to which the function can be applied. ENDI
STARTI [Basic] Question: What is the range of a function? Back: The range of a function is the set of all possible output values, i.e., the values generated as (x) varies through the domain. ENDI
STARTI [Basic] Question: How can you visualize a function? Back: A function can be visualized using a graph, the set of all Cartesian coordinates where (x) is in the domain (X) and (y = f(x)). ENDI
STARTI [Basic] Question: What is the vertical line test? Back: The vertical line test is a method to determine if a graph represents a function. If any vertical line intersects the graph more than once, then the graph does not represent a function. ENDI
STARTI [Basic] Question: What is a polynomial function? Back: A polynomial in (x) of degree (n) has the form (f(x)=a_n x^n + \ldots + a_2 x^2 + a_1 x + a_0). The domain of a polynomial expressed in this way is always (\mathbb{R}). ENDI
STARTI [Basic] Question: What is a rational function? Back: Rational functions are functions that can be expressed as the ratio of two polynomial functions. A rational function is a function (f) that can be written in the form (f(x) = \frac{P(x)}{Q(x)}) where (P(x)) and (Q(x)) are polynomial functions, and (Q(x) \neq 0). ENDI
STARTI [Basic] Question: What is a periodic function? Back: A function is periodic if there is a number, (p>0 : f(x+p)=f(x)), for all (x) in the domain of (f). The smallest such number is called the period of (f). ENDI
STARTI [Basic] Question: What are the basic trigonometric functions? Back: The basic trigonometric functions are (\sin) and (\cos). Both have the domain ((-∞,∞)) and range ([-1,1]). ENDI
STARTI [Basic] Question: What are even and odd functions? Back: An even function has reflective symmetry: (f(-x)=f(x)). An odd function has rotational symmetry: (f(-x)=-f(x)). ENDI
STARTI [Basic] Question: What are exponential functions? Back: Exponential functions are functions in which the variable is in the exponent. The general form of an exponential function is (f(x) = a \cdot b^x) where (a) and (b) are constants, and (b > 0). ENDI
STARTI [Basic] Question: What are piece-wise defined functions? Back: Piece-wise defined functions are functions that are defined by different formulas over different intervals of their domain. ENDI
STARTI [Basic] Question: What are hyperbolic functions? Back: Hyperbolic functions are analogs of the trigonometric functions for the hyperbola. Examples include (\sinh(x)), (\cosh(x)), and (\tanh(x)). ENDI
STARTI [Basic] Question: What is the composition of two functions? Back: Given two functions (f) and (g), the composition of (f) with (g) is defined as ((f \circ g)(x) = f(g(x))). ENDI
STARTI [Basic] Question: What are complex numbers? Back: Complex numbers are an extension of the real numbers and can be represented in the form (a + bi), where (a) and (b) are real numbers, and (i) is the imaginary unit with the property that (i^2 = -1). ENDI
STARTI [Basic] Question: What does the (i) in complex numbers stand for? Back: (i) is the imaginary unit with the property that (i^2 = -1). Note that sometimes (j) is used instead of (i), especially in physics/engineering. ENDI
STARTI [Basic] Question: How can you determine the real and imaginary parts of a complex number? Back: For a complex number in the form (z=a+bi), (a) is the real part, often denoted as (\Re(z)), and (b) is the imaginary part, often denoted as (\Im(z)). ENDI
STARTI [Basic] Question: How do you add and subtract complex numbers? Back: Given two complex numbers (z_1 = a + bi) and (z_2 = c + di), their sum and difference are: (z_1 + z_2 = (a + c) + (b + d)i) and (z_1 - z_2 = (a - c) + (b - d)i). ENDI
STARTI [Basic] Question: How do you multiply complex numbers? Back: The product of two complex numbers (z_1 = a + bi) and (z_2 = c + di) is (z_1 \times z_2 = (ac - bd) + (ad + bc)i). ENDI
STARTI [Basic] Question: How do you divide complex numbers? Back: The quotient of two complex numbers (z_1 = a + bi) and (z_2 = c + di) is (\frac{z_1}{z_2} = \frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{c^2 + d^2}). ENDI
STARTI [Basic] Question: What is the geometric representation of complex numbers? Back: Complex numbers can be represented geometrically on the complex plane, with the real part on the x-axis and the imaginary part on the y-axis, using Argand diagrams. ENDI
STARTI [Basic] Question: How do you compute the modulus of a complex number? Back: The modulus of a complex number (z = a + bi) is (|z| = \sqrt{a^2 + b^2}). ENDI
STARTI [Basic] Question: How is the argument (or angle) of a complex number defined? Back: The argument of a complex number (z = a + bi) is the angle (\theta) such that (\tan(\theta) = \frac{b}{a}). ENDI
STARTI [Basic] Question: What is the polar form of a complex number? Back: A complex number can be represented in polar form as (z = r(\cos(\theta) + i\sin(\theta))) where (r = |z|) is the modulus and (\theta) is the argument. ENDI
STARTI [Basic] Question: What is Euler’s formula? Back: Euler’s formula relates the exponential function to the trigonometric functions and is expressed as (e^{i\theta} = \cos(\theta) + i\sin(\theta)). ENDI
STARTI [Basic] Question: What is Euler’s identity? Back: Euler’s identity is a special case of Euler’s formula when (\theta = \pi), and is given by (e^{i\pi} + 1 = 0). ENDI
STARTI [Basic] Question: How do you multiply and divide complex numbers in exponential form? Back: Multiplication: (re^{i\theta} \times se^{i\phi} = rs e^{i(\theta + \phi)}). Division: (\frac{re^{i\theta}}{se^{i\phi}} = \frac{r}{s} e^{i(\theta - \phi)}). ENDI
STARTI [Basic] Question: What is the conjugate of a complex number? Back: The conjugate of a complex number (z = a + bi) is (\bar{z} = a - bi). ENDI
STARTI [Basic] Question: How is the relationship between complex numbers and trigonometric functions explained using Euler’s formula? Back: Euler’s formula provides a connection between complex numbers and trigonometry, stating that (e^{i\theta} = \cos(\theta) + i\sin(\theta)). This formula relates the exponential function to the trigonometric functions. ENDI
STARTI [Basic] Question: What is De Moivre’s theorem? Back: De Moivre’s theorem provides a method to raise complex numbers to integer powers and is expressed as ((\cos\theta+i\sin\theta)^n=\cos n\theta+i\sin n\theta). ENDI
STARTI [Basic] Question: Why is the Argand diagram important in understanding complex numbers? Back: The Argand diagram, a graphical representation of complex numbers, represents the real part on the x-axis and the imaginary part on the y-axis. It helps visualize the magnitude (modulus) and direction (argument) of complex numbers. ENDI
STARTI [Basic] Question: How are complex numbers related to rotation in the complex plane? Back: Multiplying complex numbers in exponential form involves adding their arguments, which corresponds to rotating one complex number by the angle of the other in the complex plane. ENDI
STARTI [Basic] Question: How are the roots of complex numbers determined? Back: The nth root of a complex number is the number which, when raised to the power of n, gives the original number. The argument of the nth root is (\frac{\text{arg}(w) + 2k\pi}{n}) for (k = 0, 1, \ldots, n-1). ENDI
STARTI [Basic] Question: What is the intuitive idea behind continuity in mathematics? Back: Continuity is when a function’s graph can be sketched without pausing the pen. If there are any stops due to gaps or jumps, then the function is discontinuous. ENDI
STARTI [Basic] Question: What are the three criteria for a function to be continuous at a point (c)? Back: 1. (f(c)) is defined. 2. The limit of (f) as (x) approaches (c) exists. 3. (\lim_{x \to c} f(x) = f(c)). ENDI
STARTI [Basic] Question: Is the function (f(x) = x^2) continuous over all real numbers? Why? Back: Yes, it is continuous because for every (x) in (\mathbb{R}), the function value and limit as (x) approaches any point are the same. ENDI
STARTI [Basic] Question: Why is the function (g(x) = \frac{1}{x}) discontinuous at (x = 0)? Back: As (x) approaches 0 from the left, (g(x)) approaches negative infinity, and from the right, it approaches positive infinity. Hence, the limit does not exist at (x = 0). ENDI
STARTI [Basic] Question: What is a removable discontinuity with an example? Back: A removable discontinuity is when a function isn’t defined at a point, but it can be made continuous with some modification. Example: (f(x) = \frac{x^2 - 1}{x - 1}) has a removable discontinuity at (x = 1). ENDI
STARTI [Basic] Question: Describe a jump discontinuity with an example. Back: Jump discontinuity is when a function jumps from one value to another at a certain point. Example: The Heaviside step function which jumps from 0 to 1 at (x = 0). ENDI
STARTI [Basic] Question: What is an oscillating discontinuity with an example? Back: Oscillating discontinuity is when a function oscillates infinitely as it approaches a point. Example: (h(x) = \sin\left(\frac{1}{x}\right)) for (x \neq 0) has an oscillating discontinuity as (x) approaches 0. ENDI
STARTI [Basic] Question: Why is continuity important in calculus? Back: Continuity is crucial in calculus because many operations, like differentiation and integration, require understanding continuity. For instance, the Fundamental Theorem of Calculus requires the integrand to be continuous over the interval of integration. ENDI
STARTI [Basic] Question: In conclusion, why is continuity essential in mathematics? Back: Continuity ensures predictable behavior of functions and serves as a prerequisite for many theorems and operations. Visualization can assist in understanding these concepts. ENDI
STARTI [Basic] Question: What is the composite function of two functions ( f ) and ( g )? Back: ( (f \circ g)(x) = f(g(x)) ). ENDI
STARTI [Basic] Question: For ( f \circ g ) to be continuous at ( x = c ), what two conditions must be satisfied? Back: ( g ) must be continuous at ( c ) and ( f ) must be continuous at ( g(c) ). ENDI
STARTI [Basic] Question: For the functions ( f(x) = x^2 ) and ( g(x) = \sin(x) ), what is ( f(g(x)) )? Back: ( f(g(x)) = \sin^2(x) ) which is continuous everywhere. ENDI
STARTI [Basic] Question: For the functions ( f(x) = \frac{1}{x} ) and ( g(x) = x + 1 ), what is ( f(g(x)) )? Back: ( f(g(x)) = \frac{1}{x + 1} ) which is continuous everywhere. ENDI
STARTI [Basic] Question: For the functions ( f(x) = \sqrt{x} ) and ( g(x) = x - 2 ), what is ( f(g(x)) )? Back: ( f(g(x)) = \sqrt{x - 2} ) which is continuous for ( x \geq 2 ). ENDI
STARTI [Basic] Question: If both functions are continuous at a point, is their composite function always continuous at that point? Back: Usually, but always check the continuity of the inner function at that point and the continuity of the outer function at the value given by the inner function. ENDI
STARTI [Basic] Question: Can discontinuities in the outer function be introduced by the inner function? Back: Yes, especially if the inner function’s range includes points where the outer function is not defined or not continuous. ENDI
STARTI [Basic] Question: What is the Intermediate Value Theorem (IVT) related to? Back: The Intermediate Value Theorem (IVT) is related to continuous functions in calculus and their behavior over a closed interval. ENDI
STARTI [Basic] Question: How is the Intermediate Value Theorem (IVT) stated? Back: Let ( f ) be a function that is continuous on the closed interval ([a, b]). If ( k ) is any value between ( f(a) ) and ( f(b) ) (exclusive), then there exists at least one ( c ) in the open interval ((a, b)) such that ( f(c) = k ). ENDI
STARTI [Basic] Question: What is the simpler interpretation of IVT? Back: If you have a continuous function that starts below a value ( k ) and ends above ( k ) (or vice versa) on an interval, then the function must take on the value ( k ) somewhere in that interval. ENDI
STARTI [Basic] Question: How can IVT be visualized? Back: Imagine drawing the graph of a continuous function over an interval without lifting your pen. If one end is below the horizontal line ( y = k ) and the other end is above it, your pen must cross the line ( y = k ) at some point in the interval. ENDI
STARTI [Basic] Question: Give an example using the function ( f(x) = x^3 - x ). Back: On the interval ([-1, 1]), ( f(-1) = -2 ) and ( f(1) = 0 ). If you pick any value between -2 and 0, say ( k = -1 ), by IVT, there exists some ( c ) in the interval ([-1, 1]) such that ( f(c) = -1 ). ENDI
STARTI [Basic] Question: Give an example using the function ( g(x) = \sin(x) ). Back: Over the interval ([0, \pi]), ( g(0) = 0 ) and ( g(\pi) = 0 ). But for any ( k ) value between 0 (exclusive) and 1, IVT guarantees a ( c ) in the interval where ( g(c) = k ). ENDI
STARTI [Basic] Question: How is the IVT useful in root-finding algorithms? Back: The IVT is used to determine that if a continuous function changes sign over an interval, then it has a root in that interval. Algorithms like the bisection method utilize this property to approximate roots. ENDI
STARTI [Basic] Question: Why is the Intermediate Value Theorem important in analysis and calculus? Back: The IVT is a foundational tool in analysis and calculus that links continuity and the behavior of functions over intervals, ensuring that continuous functions exhibit predictable behaviors, making them amenable to various mathematical and computational techniques. ENDI
STARTI [Basic] Question: What is the definition of differentiation? Back: Differentiation is a fundamental concept in calculus dealing with the rate at which a function changes. The derivative of a function provides the slope of the tangent line to the curve of that function at any given point. ENDI
STARTI [Basic] Question: How is the derivative of a function at a point defined using the limit definition? Back: The derivative of a function at a point is defined as the limit of the average rate of change of the function over a small interval around that point as the interval’s width approaches zero, formally given by (f’(x) = \lim_{{h \to 0}} \frac{f(x + h) - f(x)}{h}). ENDI
STARTI [Basic] Question: How is the derivative of the function (f(x) = x^2) found using the limit definition? Back: The derivative is found as follows: (f’(x) = \lim_{{h \to 0}} \frac{(x+h)^2 - x^2}{h} = \lim_{{h \to 0}} (2x + h) = 2x). ENDI
STARTI [Basic] Question: What is the Constant Rule in differentiation? Back: The Constant Rule states that the derivative of a constant is zero, represented as (\frac{d}{dx} c = 0), where (c) is a constant. ENDI
STARTI [Basic] Question: What is the Power Rule in differentiation? Back: The Power Rule states that for any real number (n), (\frac{d}{dx} x^n = nx^{n-1}). ENDI
STARTI [Basic] Question: What is the Product Rule in differentiation? Back: The Product Rule states that for functions (f(x)) and (g(x)), (\frac{d}{dx} [f(x) \cdot g(x)] = f’(x)g(x) + f(x)g’(x)). ENDI
STARTI [Basic] Question: Using the Product Rule, how would you differentiate (y = x^2 \sin(x))? Back: Differentiate (y) as follows: [\frac{d}{dx} (x^2 \sin(x)) = x^2 \cdot \frac{d}{dx}(\sin(x)) + \sin(x) \cdot \frac{d}{dx}(x^2) = x^2 \cdot \cos(x) + 2x \cdot \sin(x)]. ENDI
STARTI [Basic] Question: How do you find the equation of the tangent line to the curve (y = x^2) at the point ((1,1))? Back: The derivative at (x = 1) is (2(1) = 2). So, the slope of the tangent is (2). Using the point-slope form (y - y_1 = m(x - x_1)), the equation of the tangent line is (y - 1 = 2(x - 1)) or (y = 2x). ENDI
STARTI [Basic] Question: What does the derivative (\frac{d}{dx} x) equal to? Back: The derivative (\frac{d}{dx} x) equals to (1). ENDI
STARTI [Basic] Question: How is the derivative of the function (f(x) = x^2) at a general point (x) denoted? Back: The derivative is denoted as (f’(x)) or (\frac{df}{dx}), and is equal to (2x). ENDI
STARTI [Basic] Question: What is the formula for Logarithmic Differentiation? Back: Begin by writing (y=f(x)), then take natural logarithms of both sides and simplify (\ln(f(x))) using the laws of logarithms. Differentiate both sides of the equation with respect to (x), then solve for (\frac{dy}{dx}). ENDI
STARTI [Basic] Question: Differentiate the function (y = 3x^2 - 4x + 7). Back: The derivative is found as (y’ = 6x - 4). ENDI
STARTI [Basic] Question: What is the Product Rule in logarithms? Back: The Product Rule in logarithms is given by (\ln(ab) = \ln a + \ln b). ENDI
STARTI [Basic] Question: What is the Quotient Rule in logarithms? Back: The Quotient Rule in logarithms is given by (\ln\left(\frac{a}{b}\right) = \ln a - \ln b). ENDI
STARTI [Basic] Question: What is the Exponent Rule in logarithms? Back: The Exponent Rule in logarithms is given by (\ln(a^r) = r\ln a). ENDI
STARTI [Basic] Question: How do you use implicit differentiation to find (\frac{dy}{dx}) for the equation (x^2 + y^2 = 25)? Back: To find (\frac{dy}{dx}), differentiate both sides of the equation to get (2x + 2y\frac{dy}{dx} = 0). Then, solve for (\frac{dy}{dx}) to obtain (\frac{dy}{dx} = -\frac{x}{y}). ENDI
STARTI [Basic] Question: What is the relationship between a function ( f ) and its inverse ( f^{-1} )? Back: For a function ( f ) and its inverse ( f^{-1} ), the relationships are ( f(f^{-1}(x)) = x ) and ( f^{-1}(f(x)) = x ). The graph of an inverse function is the reflection of the original function’s graph across the line ( y = x ). ENDI
STARTI [Basic] Question: What are the inverse hyperbolic functions for ( \sinh(x) ), ( \cosh(x) ), and ( \tanh(x) )? Back: The inverse hyperbolic functions for ( \sinh(x) ), ( \cosh(x) ), and ( \tanh(x) ) are ( \text{arsinh}(x) ), ( \text{arcosh}(x) ), and ( \text{artanh}(x) ) respectively. ENDI
STARTI [Basic] Question: What are critical points of a function? Back: Critical points are points on a graph where the function has local minima, maxima, or saddle points, occurring where the derivative ( f’(x) = 0 ) or where ( f’(x) ) is undefined. ENDI
STARTI [Basic] Question: What does the Extreme Value Theorem state? Back: The Extreme Value Theorem states that if a function ( f(x) ) is continuous on a closed interval ([a,b]), then ( f(x) ) attains both a global maximum and minimum on that interval. ENDI
STARTI [Basic] Question: How does Fermat’s Theorem relate to finding local extrema? Back: Fermat’s Theorem states that if ( f(x) ) has a local extremum at ( x = c ) and ( f’(c) ) exists, then ( f’(c) = 0 ). It helps in pinpointing where local maxima and minima can occur. ENDI
STARTI [Basic] Question: What does the Mean Value Theorem guarantee? Back: The Mean Value Theorem guarantees that if ( f(x) ) is continuous on ([a, b]) and differentiable on ((a, b)), there exists at least one point ( c ) in ((a, b)) where ( f’(c) = \frac{f(b) - f(a)}{b - a} ), indicating an instantaneous rate of change equal to the average rate of change over that interval. ENDI
STARTI [Basic] Question: What is Rolle’s Theorem and how is it a special case of the Mean Value Theorem? Back: Rolle’s Theorem is a special case of the Mean Value Theorem where ( f(a) = f(b) ). It states that there exists at least one point ( c ) in ( (a, b) ) such that ( f’(c) = 0 ), indicating a horizontal tangent exists at some point when the function values at the endpoints are equal. ENDI
STARTI [Basic] Question: How do you determine intervals of increase and decrease for a function? Back: To find where the function is increasing or decreasing, find the critical points by solving ( f’(x) = 0 ). Classify these points into intervals and pick a test point within each interval to determine the sign of ( f’(x) ). ENDI
STARTI [Basic] Question: How can you classify a critical point as a local maximum or minimum? Back: A critical point is a local maximum if ( f’(x) > 0 ) before and ( f’(x) < 0 ) after it. It’s a local minimum if ( f’(x) < 0 ) before and ( f’(x) > 0 ) after the critical point. ENDI
STARTI [Basic] Question: What does the concavity of a function tell you and how do you test for it? Back: Concavity indicates the “direction” of a curve. A function is concave up where ( f”(x) > 0 ) and concave down where ( f”(x) < 0 ). To test for concavity, you check the sign of the second derivative ( f”(x) ). ENDI
STARTI [Basic] Question: What is a point of inflection and its significance in graph sketching? Back: A point of inflection is where the concavity of the function changes, and is identified where ( f”(x) = 0 ) or is undefined. These points are important in graph sketching as they indicate a change in the “bending direction” of the graph. ENDI
STARTI [Basic] Question

Weeks Content

The Knowledge Cottage

Table of Contents

MTH1002 Calculus

Calculus

Course Components

Outline Syllabus

Learning Outcomes

Flashcards

Weeks Content

Week 1

Week 2

Week 3

Week 4

Week 5

Week 6

Week 8

Graph View

Backlinks

The Knowledge Cottage

Table of Contents

MTH1002 Calculus

Calculus §

Course Components §

Outline Syllabus §

Learning Outcomes §

Flashcards §

Weeks Content §

Week 1 §

Week 2 §

Week 3 §

Week 4 §

Week 5 §

Week 6 §

Week 8 §

Graph View

Backlinks

Calculus

Course Components

Outline Syllabus

Learning Outcomes

Flashcards

Weeks Content

Week 1

Week 2

Week 3

Week 4

Week 5

Week 6

Week 8