MTH-112 Calculus 1: Limits, Derivatives, Integrals

• MTH-112 Calculus 1 is taught in Spring 2024, Lecture 1 by Dr. Ahmed Makki

Textbook

• The textbook is Thomas' Calculus, 14th edition SI units by Thomas, Weir, and Hass, published by Pearson Education in Boston in 2019
• The textbook's ISBN is 9781292253220

Assessment

• Quizzes are worth 30% of the final grade
• The midterm examination is worth 30% of the final grade
• The final examination is worth 40% of the final grade

Course Description

• Differential calculus topics include limits, continuity, higher-order derivatives, curve sketching, and differentials
• Integral calculus topics include definite and indefinite integrals (areas and volumes), and applications of derivatives and integrals
• Students will develop skills to apply calculus concepts to solve problems in science and engineering

Learning Outcomes

• Upon completion of this course, students will be able to compute limits and derivatives of functions
• Students will be able to formulate and solve time-related and optimization problems in engineering and computing
• Students will be able to evaluate integrals and calculate areas and apply symbolic math software to calculate derivatives and integrals

Course Timeline:

• Week 1: Functions
• Week 2: Calculating Limits Using the Limit Laws and One-Sided Limits and Limits at Infinity
• Week 3: Infinite Limits and Vertical Asymptotes and Differentiation Rules, Quiz 1
• Week 4: The Derivative as a Rate of Change
• Week 5: Derivatives of Trigonometric Functions and Derivative and the Slope
• Week 6: The Chain Rule and Implicit Differentiation, Quiz 2
• Week 7: Related Rates and Derivatives using symbolic math software
• Week 8: Extreme Values of Functions, Midterm
• Week 9: Monotonic Functions and the First Derivative Test, Concavity and Curve Sketching
• Week 10: Applied Optimization Problems
• Week 11: Indeterminate Forms and L’Hôpital's Rule Anti-derivatives
• Week 12: Estimating with Finite Sums and The Definite Integral
• Week 13: The Fundamental Theorem of Calculus and Indefinite Integrals and the Substitution Rule
• Week 14: Substitution and Area Between Curves and Integrals using symbolic math software, Quiz 4
• Week 15: Integrals using symbolic math software Review
• Week 16: Final exam

Quick Review

• To find the value of y that corresponds to x=3 in the equation y = -2 + 4(x - 3), substitute x=3 into the equation
• To find the value of x that corresponds to y=3 in the equation y = 3 - 2(x + 1), substitute y=3 into the equation
• To find the value of m that corresponds to the values of x and y, substitute the values of x and y into the equation for example
• x=5, y =2, m = (y-3)/(x-4)

Quick Review Solutions

• The value of y when x=3 in y = -2 + 4(x - 3) equals -2
• The value of x when y=3 in y = 3 - 2(x + 1) equals -1
• When x=5 and y=2, m = -1
• When x=-1 and y=-3, m = 5/4
• In the equation 3x - 4y = 5, the ordered pair (2, 1/4) is a solution
• In the equation y = -2x + 5, the ordered pair (-1, 7) is a solution
• The distance between the points (1,0) and (0,1) is √2
• The distance between the points (2,1) and (1, -1/3) is 5/3
• Solving 4x - 3y = 7 for y in terms of x gives y = (4/3)x - (7/3)
• Solving -2x + 5y = -3 for y in terms of x gives y = (2/5)x - (3/5)

Increments

• When a particle moves from point (x₁, y₁) to point (x₂, y₂), the increments in its coordinates are Δx = x₂ - x₁ and Δy = y₂ - y₁
• Δx and Δy are read as "delta x" and "delta y," representing the change in x and y
• Δ (delta) is a Greek capital letter that denotes difference

Sample Increments

• The coordinate increments from (8, 3) to (-6, 1) are Δx = -6 - 8 = -14 and Δy = 1 - 3 = -2

Slope of a Line

• The slope (m) of a nonvertical line L, passing through points P₁(x₁, y₁) and P₂(x₂, y₂), is defined as m = (rise) / (run) = (y₂ - y₁) / (x₂ - x₁)
• A line that goes uphill as x increases has a positive slope, and a line that goes downhill as x increases has a negative slope
• A horizontal line has a slope of zero, while vertical lines have undefined slope

Equations of Lines

• The vertical line through the point (a, b) has the equation x = a
• The horizontal line through the point (a, b) has the equation y = b

Sample Equations of Lines

• The vertical and horizontal lines through the point (-3, 8) are x = -3 and y = 8

Point Slope Equation

• The point-slope equation of a line through point (x₁, y₁) with slope m is given by y = m(x - x₁) + y₁

Example: Point Slope Equation

• The point-slope equation for the line through (7, -2) and (-5, 8) is y = (-5/6)x + (23/6)

Section 1.2

• This section introduces functions and their graphs

Function

• A function from set D to set R is a rule that assigns a unique element in R to each element in D
• D is the domain and R contains the range

Example Functions

• With f(x) = 2x + 3, when x=6, f(6)=15

Domains and Ranges

• Domains and ranges of real-valued functions are intervals, which can be open, closed, half-open, finite, or infinite
• Endpoints are boundary points
• Remaining points are interior points
• Closed intervals contain boundary points
• Open intervals contain no boundary points

Graph

• A graph consists of the points (x, y) in the plane, where these coordinates are the input-output pairs of a function y = f(x)
• y = x² has a Domain of (-∞, ∞) and a Range of [0, ∞)
• y = 1/x has a Domain of (-∞, 0) U (0, ∞) and a Range of (-∞, 0) U (0, ∞)
• y = √x has a Domain of [0, ∞) and a Range of [0, ∞)
• y = √(4 - x) has a Domain of (-∞, 4] and a Range of [0, ∞)
• y = √(1 - x²) has a Domain of [-1, 1] and a Range of [0, 1]

Functions Defined in Pieces

• Piecewise functions are defined by applying different formulas to different parts of their domain

Absolute Value Functions

• The absolute value function y = |x| is defined piecewise: being -x for x < 0 and x for x ≥ 0
• The absolute value function is even, with a graph symmetrical about the y-axis

Composite Functions

• If the outputs of a function g can be inputs of a function f, then g and f form a composite function, with inputs x and outputs f(g(x))
• The composite of g and f is written f o g

Example Composite Functions

• Given f(x) = 2x - 3 and g(x) = 5x, the composite function (f o g)(x) = 10x - 3

Exponential Function

• For a positive real number a not equal to 1, f(x) = a^x is an exponential function with base a
• The domain of f(x) = a^x is (-∞, ∞) and the range is (0, ∞)
• Compound interest investment and population growth are examples of exponential growth

Exponential Growth

• If a > 1, the graph of f(x) resembles the graph of y = 2^x

Exponential Growth

• If 0 < a < 1, the graph of f(x) resembles the graph of y = 2^(-x)

Rules for Exponents

• If a > 0 and b > 0, the following rules hold for all real numbers x and y:
• a^x * a^y = a^(x+y)
• a^x/a^y = a^(x-y)
• (a^x)^y = (a^y)^x = a^(xy)
• a^x * b^x = (ab)^x
• (a/b)^x = a^x/b^x

Exponential Growth and Exponential Decay

• A function y = k*a^x, where k > 0, models exponential growth if a > 1 and exponential decay if 0 < a < 1

Example Exponential Functions

• The zero of f(x) = 4^x - 3 is x ≈ 0.79248125

The Number e

• Many natural, physical, and economic phenomena are modeled by an exponential function whose base is the number e, approximately 2.718281828
• The definition of e is the value that the function f(x) = (1 + 1/x)^x approaches as x approaches infinity
• y = e^x and y = e^(-x) are frequently used as models of exponential growth or decay
• The model y = P*e^(rt) is used to calculate interest compounded continuously, with P as starting principal, r as interest rate, and t as time in years

Inverse Properties for a^x and logₐ x

• a^(logₐ x) = x, where a > 1 and x > 0
• e^(ln(x)) = x and ln(e^x) = x, where x > 0

Example Properties of Logarithms

• Solving 2^x = 12 involves taking the natural logarithm of both sides, allowing x to be isolated. The solution is approximately x ≈ 3.32193
• To solve e^x + 5 = 60, the natural logarithm is taken, meaning x ≈ 4.007333

Trigonometric Functions

• cos²θ + sin²θ = 1
• 1 + tan²θ = sec²θ
• 1 + cot²θ = csc²θ
• cos(A + B) = cosAcosB - sinAsinB
• sin(A + B) = sinAcosB + cosAsinB
• cos2θ = cos²θ - sin²θ
• sin2θ = 2sinθcosθ
• cos²θ = (1 + cos2θ)/2
• sin²θ = (1 - cos2θ)/2

Radian Measure

• For an angle of measure θ in standard position at the center of circle of radius r, y represents the point P(x,y) on its terminal rate

Trigonometric Functions of θ

• sine: sin θ= y/r
• cosine: cos θ= x/r
• tangent: tan θ = y/x
• cosecant: csc θ = r/y
• secant: sec θ =-r/x
• cotangent: cot θ = x/y