MATH 1760 - Analytic Geometry & Calculus 1

Credit Hours: 4.00

Objectives: Students will:

1. Examine a data table for a function and make reasonable conjectures about limits.
2. Examine the graph of a function and make reasonable conjectures about limits.
3. Recognize indeterminate forms 0/0 and infinity/infinity.
4. Find limits of appropriate functions using the rules of limits.
5. Use limits to determine asymptotes of appropriate functions.
6. Recognize indeterminate forms 0/0 and infinity/infinity, 0*infinity, infinity-infinity, 0^0, 1^infinity, infinity^0 and find limits using L’Hopital’s Rule.

Outcome 2: Upon successful completion of this course, the student will be able to analyze the continuity of a function graphically and analytically.

Objectives: Students will:

1. Examine a function graphically and make reasonable conjectures about continuity at a point.
2. Use the definition to determine whether a function is continuous at a point.
3. Determine the intervals on which a function is continuous.
4. Classify continuities using limits to justify.

Outcome 3: Upon successful completion of this course, the student will be able to calculate the derivative of a function numerically and analytically.

Objectives: Students will:

1. Know the definition of a derivative.
2. Use the definition of a derivative to approximate a derivative numerically.
3. Use the definition of a derivative to find the derivative of an algebraic function.
4. Use the rules and formulas to differentiate appropriate functions.
5. Differentiate exponential and logarithmic functions of any base.
6. Differentiate trigonometric and inverse trigonometric functions.
7. Find the derivative of an implicitly defined function.
8. Use logarithmic differentiation to differentiate functions of the form f(x)^g(x).

Outcome 4: Upon successful completion of this course, the student will be able to calculate antiderivatives and to use them to solve basic differential equations.

Objectives: Students will:

1. Know the relationship between a derivative and an antiderivative.
2. Find the antiderivative of basic functions by rules and substitutions.
3. Solve simple differential equations.

Outcome 5: Upon successful completion of this course, the student will be able to calculate a definite integral numerically and analytically.

Objectives: Students will:

1. Approximate definite integrals by appropriate numerical methods.
2. Find the definite integral of appropriate functions by the Fundamental Theorem of Calculus.

Outcome 6: Upon successful completion of this course, the student will be able to use derivatives obtain tangent lines and to deduce detailed information about the shape of a function’s graph.

Objectives: Students will:

1. Find the equation of the tangent line to the graph of a function at a given point.
2. Write the equation of a tangent line to an implicitly defined function.
3. Determine intervals of increasing and decreasing behavior of appropriate functions.
4. Determine extrema of appropriate functions.
5. Determine concavity of appropriate functions.
6. Sketch the graph of a function using the first and second derivative.

Outcome 7: Upon successful completion of this course, the student will be able to use derivatives to solve appropriate applications.

Objectives: Students will:

1. Use derivatives to solve applications involving extrema.
2. Use derivatives to solve applications involving related rates of change.

Outcome 8: Upon successful completion of this course, the student will be able to use definite integrals to calculate areas.

Objectives: Students will:

1. Use a definite integral to find the area between the graph of a function and the x-axis.
2. Use a definite integral to find the area between the graphs of functions.

1. Review
   1. Lines
   2. Functions and graphs
2. Concepts
   1. Average rate of change
   2. Secant lines
   3. Instantaneous rate of change
   4. Tangent lines
   5. Limits, limits at infinity & infinite limits
   6. The derivative
3. Differentiation
   1. Constants and the power rule
   2. Product & quotient rules
   3. Trigonometric functions
   4. Logarithmic and exponential functions
   5. Inverse trigonometric functions
   6. Composite functions and the chain rule
   7. Implicit differentiation
4. The Derivative and Application
   1. Rates of change
   2. Slope of a tangent line
   3. Related rates
   4. The shape of a graph
   5. The Mean Value Theorem
   6. Optimization
   7. Differentials
   8. Indeterminate forms and L’Hopital’s rule
5. Integration
   1. Indefinite integrals
   2. Integration Rules
   3. Integration by substitution
   4. Differential equations
   5. Riemann sums and the definite integral
   6. The Mean Value and Fundamental Theorems
   7. Numerical Integration
   8. Area between curves