# MA1113: Single Variable Calculus I - NPS Online

# Single Variable Calculus I

Course #MA1113

**Est.imated Completion Time:**
3
months

**POC:**
NPS Online Support

## Overview

Review of analytic geometry and trigonometry, functions of one variable, limits, derivatives, continuity and differentiability; differentiation of algebraic, trigonometric, logarithmic and exponential functions with applications to maxima and minima, rates, differentials; product rule, quotient rule, chain rule; anti-derivatives, integrals and the fundamental theorem of calculus; definite integrals, areas. Taught at the rate of nine hours per week for five weeks. Prerequisites: None.

## Learning Outcomes

Functions and limits:

- Use real numbers, inequalities involving real numbers and their absolute values, the trigonometric functions, and the radian measure of angles.
- Be able to move back and forth between the descriptions of a function by an equation, a table, a graph, and by words.
- Be able to use exponential functions, sketch their graphs, and define the number e.
- Define what it means for a function to be one-to-one and determine whether a function has an inverse or not and sketch its inverse if it does.
- Be able to use logarithmic functions, sketch their graphs, and define the relationship between the natural exponential and natural logarithmic functions.
- State in words what it means for a function to have a limit, be able to calculate limits, and be able to find the vertical and horizontal asymptotes of a function.
- State in words what it means for a function to be continuous and be able to find limits for continuous functions.

Derivatives:

- Relate the notions of tangent to a curve, velocity, and rate of change, and illustrate them in a sketch.
- State the definition of derivative as the limit of a difference quotient and explain how the derivative itself can be regarded as a function.
- Be able to find derivatives of polynomials and exponential functions.
- State the product and quotient rules for differentiation and be able to use them to differentiate functions.
- Know the derivatives of sine and cosine and be able to use the quotient rule to determine the derivatives of the remaining four trigonometric functions.
- State the chain rule and use it to differentiate functions obtained by composition.
- Use the differentiation rules to differentiate implicitly, and to find higher order derivatives.
- Be able to differentiate logarithmic functions, and functions involving them.
- Define the hyperbolic functions and be able to differentiate them.

Applications of Differentiation:

- Be able to solve related rates problems. Understand them as an application of the chain rule.
- Understand the connection between the derivative, the tangent line to the graph of a function, the linearization of a function, and the differential of a function.
- Use the differential (or linearization) to solve “small change” and applied approximation problems.
- Be able to state the Mean Value Theorem and give some of its consequences.
- Describe how the signs of the first and second derivatives of a function affect the shape of its graph.
- Define and recognize the various forms of indeterminate forms, and use L”Hospital’s Rule to determine their limits.
- Be able to set up and solve optimization problems using calculus methods.
- Be able to describe Newton’s method geometrically, and to use it to iteratively approximate the zeros of functions.
- Define what the antiderivative of a function is and be able to find it for reasonable functions.

Integral Calculus:

- Describe the connection between the problems of finding areas and distances travelled, and how both problems lead to the same limit.
- Know and be able to work with the properties of definite integrals.
- State the Fundamental Theorem of Calculus in words, describe how it connects integral and differential calculus, and how it helps in finding antiderivatives and in evaluating definite integrals.
- Define the indefinite integral of a function and state its relation to the antiderivative.
- Be able to use the Substitution Rule to evaluate definite and indefinite integrals.
- Be able to use integration by parts to evaluate appropriate integrals.

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