MTH
175 Syllabus

**Division**: Arts and
Sciences Date: February 2014

**Curricula in Which Course
is Taught**: Science,
Liberal Arts

**Course Number and Title**: MTH
175, Calculus of One Variable I

**Credit Hours**: 3 **Hours/Wk
Lecture**: 3 **Hours/Wk Lab**: **Lec****/Lab Comb**: 3

I.
**Catalog
Description: **Presents
differential calculus of one variable including the theory of limits,
derivatives, differentials, antiderivatives and
applications to algebraic and transcendental functions. Designed for mathematical,
physical, and engineering science programs.

II.
**Relationship of
the course to curricula objectives in which it is taught**: This course allows a student to use logical and
mathematical reasoning in problem solving in theoretical and applied areas that
require critical thinking.

III.
**Required background**: A placement recommendation for MTH 175 and
four units of high school mathematics including Algebra I, Algebra II, Geometry
and Trigonometry or equivalent. Non-developmental through MTE 9.

IV.
**Course Content**:

A.
Graphs

B.
Linear Models and Rates of Change

C.
Functions and Their Graphs

D.
Fitting Models to Data

E.
Inverse Functions

F.
Exponential and Logarithmic
Functions

G.
A Preview of Calculus & Tangent
line problem

H.
Finding Limits Graphically and
Numerically

I.
Evaluating Limits Analytically

J.
Continuity and One-Sided Limits

K.
Infinite Limits

L.
The Derivative and the Tangent Line
Problem

M.
Basic Differentiation Rules and
Rates of Change

N.
The Product and Quotient Rules and
Higher-Order Derivatives

O.
The Chain Rule

P.
Implicit Differentiation

Q.
Derivatives of Inverse Functions

R.
Related Rates

S.
Newton’s Method

T.
Extrema on an Interval

U.
Rolle’s Theorem and the Mean Value
Theorem

V.
Increasing and Decreasing Functions
and the First Derivative Test

W.
Concavity and the Second Derivative Test

X.
Limits at Infinity

Y.
A Summary of Curve Sketching

Z.
Optimization Problems

AA.
Differentials

V.
Upon
completion of the course the students will be
able to: 1.
Demonstrate the ability to sketch and recognize basic algebraic
and transcendental functions. 2.
Sketch the graph of one or more equations and state
graphical information displayed. 3.
Write equations of lines given particular conditions. 4.
Interpret slope as a ratio or as a rate in a real-life
application. 5.
Use function notation to represent and evaluate a
function stating its domain and range using proper interval notation and
apply to real life situations. 6.
Identify different types of transformations of
functions. 7.
Classify functions, recognize combinations of functions,
and evaluate composite functions. 8.
Interpret mathematical models for real-life data and
fit a linear, quadratic, or trigonometric model to real-life data. 9.
Demonstrate the ability to solve higher level equations
and absolute value equations. 10.
Demonstrate an understanding of what calculus is about,
how it compares to precalculus, and how both the
tangent line and area problem are basic to calculus. 11.
Estimate a limit using a numerical or graphical
approach. 12.
Know different ways a limit can fail to exist. 13.
Know and use the formal definition of a limit; complete
a _ 14.
Evaluate a limit analytically by applying the
properties of limits, dividing out and rationalizing techniques, and the
Squeeze Theorem. 15.
Develop and use a strategy for finding limits. 16.
Demonstrate an understanding of continuity at a point,
on an open interval, or on a closed interval, and apply the properties of
continuity. 17.
Determine and evaluate one-sided limits. 18.
Understand and use the Intermediate Value Theorem. 19.
Determine infinite limits from the left and from the
right; apply infinite limit properties. 20.
Find and sketch the vertical asymptotes of the graph of
a function. 21.
Find the slope of the tangent line to a curve at a
point. 22.
Use the limit definition to find the derivative of a
function. 23.
Understand the relationship between differentiability
and continuity. 24.
Apply the Constant Rule, Power Rule, the Constant
Multiple Rule, and/or the Sum and Difference Rules to find the derivative of
a function. 25.
Find the derivatives of the sine and cosine functions. 26.
Use derivatives to find rates of change. Understand the
velocity and position functions. 27.
Apply the Product Rule and Quotient Rule to find the
derivative of a function. 28.
Find higher-order derivatives of a function. 29.
Understand and apply the Chain Rule to find the
derivative of a composite and trigonometric function. 30.
Find the derivative of a function using the General
Power Rule. 31.
Simplify the derivative of a function using algebra. 32.
Ability to distinguish between functions written in
implicit and explicit form. 33.
Use implicit differentiation to find the derivative of
a function. 34.
Find related rates and use related rates to solve
real-life problems. 35.
Understand the definition of extrema
of a function on an interval and relative extrema
on an open interval. 36.
Find the extrema on a closed
interval. 37.
Understand and use both Rolle’s Theorem and the Mean
Value Theorem. 38.
Determine intervals on which a function is increasing
or decreasing. 39.
Apply the First Derivative Test to find relative extrema of a function. 40.
Determine intervals on which a function is concave
upward or concave downward. 41.
Find any points of inflection of the graph of a
function. 42.
Apply the Second Derivative Test to find relative extrema of a function. 43.
Determine finite and infinite limits at infinity and
state the horizontal asymptotes, if any, of the graph of a function. 44.
Analyze and sketch the graph of a function without the
aid of a calculator, stating all pertinent information about the sketch. 45.
Demonstrate the ability to solve applied minimum and
maximum problems (optimization). 46.
Approximate a zero of a function using Newton’s Method. 47.
Understand the concept of a tangent line approximation. 48.
Compare the value of the differential, 49.
Estimate the propagated error using a differential. 50.
Find the differential of a function using
differentiation formulas. 51.
Find the inverse of a function; derivative of an
inverse function. 52.
Demonstrate properties of logarithmic and exponential
functions; find their derivatives. |
VI. Test, quizzes, written assignments |

VII. **This course
supports the following objectives**:

**DCC Educational Objectives:**

Critical
Thinking

Information Literacy

Quantitative Reasoning

Scientific Reasoning