Calculus I MATH 150/151 Course Notes

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Questions and Answers

Which set of numbers includes only positive counting numbers?

Integers

Rational Numbers

Real Numbers

Natural Numbers (correct)

What is the definition of a function's range?

The total number of elements in the codomain.

The unique elements of set X.

The set of all elements in Y corresponding to an element of X. (correct)

The set of all possible inputs.

How can a function be represented numerically?

By a formula.

By a verbal description.

Using a graph.

By a table of values. (correct)

Which of the following statements about rational numbers is true?

They can be expressed as the ratio of two integers. Signup and view all the answers

What is the primary difference between the sets of rational and real numbers?

Real numbers include all rational numbers and irrational numbers. Signup and view all the answers

Which of the following is not a valid way to define a function?

Expressively. Signup and view all the answers

Which of the following best describes the set of integers?

All whole numbers, both positive and negative. Signup and view all the answers

Which representation of a function is described as giving a verbal description of its behavior?

Verbal representation. Signup and view all the answers

What is the velocity at time 0.6 seconds?

0.8 m/s Signup and view all the answers

Which of the following is true about the function f(x) = x^2?

f(π) = π^2 Signup and view all the answers

What would be the expression for the height y as a function of x in the ladder problem?

y = √(L^2 - x^2) Signup and view all the answers

What is the domain of the function g(x) = 1/(x^2 - x)?

x ≠ 0 and x ≠ 1 Signup and view all the answers

Which of the following accurately represents the range of the function h(t) = √(16 - t^2)?

[0, 4] Signup and view all the answers

What is the value of f(2/3) if f(x) = x^2?

4/9 Signup and view all the answers

What is the correct way to express the velocity values as a set of ordered pairs?

{(0, 0), (0.2, 0.2), (1.0, 0.6)} Signup and view all the answers

For the function f(x) = |x|, what is the graph's property?

It is symmetric about the y-axis Signup and view all the answers

What best describes the vertical line test?

It verifies if every vertical line intersects the curve at most once. Signup and view all the answers

Which of the following is a correct piecewise function for f(x) = |x|?

f(x) = x if x > 0 or f(x) = -x if x ≤ 0 Signup and view all the answers

Which of the following best describes power functions?

Functions of the form f(x) = xa where a is a fixed real number. Signup and view all the answers

In which case is the function g(x) = x^2 equal to f(x) = |x|?

When x is positive. Signup and view all the answers

How is a polynomial defined?

As a function of the form f(x) = a_n x^n + a_n-1 x^(n-1) + ... + a_1 x + a_0. Signup and view all the answers

What is the shape of the graph for the function p(x) = |x| + |x + 1|?

A V-shape with a single vertex. Signup and view all the answers

When finding the equation of a line given point P(1, 3) and slope 2, what is it?

y = 2x + 3 Signup and view all the answers

Which of the following is NOT a type of function mentioned?

Exponential functions Signup and view all the answers

What is the main focus of Part One in the Calculus I course notes?

Review of Functions and Models Signup and view all the answers

Which rule would be most appropriate for finding the derivative of a product of two functions?

Product Rule Signup and view all the answers

Which of the following is a key aspect of the Mean Value Theorem?

It guarantees at least one point where the derivative is zero. Signup and view all the answers

What type of functions are primarily reviewed in the section dedicated to Mathematical Models?

Essential Functions Signup and view all the answers

What is the primary subject of the section dedicated to Limits at Infinity?

Understanding horizontal asymptotes Signup and view all the answers

What does implicit differentiation allow a student to do?

Differentiate functions that are not explicitly solved for y. Signup and view all the answers

Which of the following best describes the Chain Rule in differentiation?

A technique applied when differentiating composite functions. Signup and view all the answers

What is one application of derivatives mentioned in the course notes?

Determining the tangent line at a point Signup and view all the answers

What differentiates polar coordinates from Cartesian coordinates?

Polar coordinates depend on angles and distances from a fixed point. Signup and view all the answers

In the context of calculus, what are related rates?

Derivatives that involve two or more quantities that change over time. Signup and view all the answers

What method can be used to solve optimization problems in calculus?

Finding critical points using derivatives. Signup and view all the answers

Which of the following derivatives specifically relate to logarithmic functions?

The property of logarithm differentiation involving the base. Signup and view all the answers

What is emphasized in the summary of curve sketching?

Understanding the influence of derivatives on graph shapes. Signup and view all the answers

The use of Newton’s Method is primarily for what purpose in calculus?

Estimating the roots of functions. Signup and view all the answers

What is the value of $ an(30^{ ext{°}})$?

$rac{ ext{√3}}{3}$ Signup and view all the answers

How does the graph of $y = f(x) + c$ relate to $y = f(x)$?

It is shifted vertically upward. Signup and view all the answers

What transformation does the equation $y = f(x - c)$ represent?

Shift to the right by $c$ units. Signup and view all the answers

What is the value of $ heta$ for which $ an( heta) = 1$?

$rac{ ext{π}}{4}$ Signup and view all the answers

Which of the following equations represents a vertical shift downward?

$y = f(x) - c$ Signup and view all the answers

For the function $y = 3x^2 - 6x + 1$, what is the effect of the $-6x$ term?

It determines the position of the vertex. Signup and view all the answers

What is the value of $ ext{sin}(90^{ ext{°}})$?

1 Signup and view all the answers

What effect does the factor of 2 in the equation $y = ext{sin}(2x)$ have on the graph of $y = ext{sin}(x)$?

It compresses the graph horizontally. Signup and view all the answers

Study Notes

Overview of Calculus I (MATH 150/151)

Course focuses on fundamental concepts of calculus, including functions, differentiation, and applications.
Offers comprehensive coverage of essential mathematical models and theories.

Functions and Models

Functions relate each element of a domain to a unique element of a codomain.
Four representations of functions:
- Verbally (e.g., descriptions)
- Algebraically (e.g., formulas)
- Numerically (e.g., tables or sets of ordered pairs)
- Visually (e.g., graphs)

Basic Sets of Numbers

Natural Numbers: N = {1, 2, 3, ...}
Integers: Z = {..., -2, -1, 0, 1, 2, ...}
Rational Numbers: Q = {a/b: a, b ∈ Z, b ≠ 0}
Real Numbers: Includes all rational numbers and irrational numbers that fill gaps within the number line.

Defining Functions

Example: For a function f such that f(x) = x², identify input-values and compute output values.
Domain represents possible input values; the codomain represents allowable output values.
Range is the actual output values corresponding to the domain.

Mathematical Models

Linear Functions: Defined by slope and y-intercept. Can deduce the equation from these parameters.
Power Functions: Functions of the form f(x) = x^a, where a is a real number.
Polynomials: Functions expressed as f(x) = a_n * x^n + ... + a_1 * x + a_0, with integer n and coefficients a_i being real numbers.

Graphing Functions

Vertical line test determines whether a graph represents a function; if any vertical line intersects at most once, it is a function.
Common function shapes include linear, parabolic, and piecewise functions.

Transformation of Functions

Transformations of graphs include:
- Shifting vertically (up/down)
- Shifting horizontally (right/left)
- Reflecting across axes
Example transformations:
- y = f(x) + c shifts upward by c.
- y = f(x) - c shifts downward by c.

Differentiation and Applications

Concepts of limits and derivatives underpin calculus.
Derivatives measure rates of change and are essential for understanding curve behavior.
Applications include optimization problems, curve sketching, and understanding motion.

Exam Preparation

Practice exercises covering all the course material are necessary.
Review notes and key concepts regularly.
Prepare sample problems and check against solutions for mastery.

Appendix and Additional Resources

Solutions to exercises provided to enhance understanding.
Bibliography includes articles, books, and web resources for further reading and study.

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Description

Explore the foundational concepts of Calculus I through these comprehensive course notes from SFU. This resource provides essential information for MATH 150/151, making complex ideas more accessible. Ideal for students needing a thorough review or supplementary material.