MAT1214

From Department of Mathematics at UTSA
Revision as of 15:23, 16 June 2020 by James.kercheville (talk | contribs) (Added additional links for prerequisites)
Jump to navigation Jump to search

The textbook for this course is Calculus (Volume 1) by Gilbert Strang, Edwin Herman, et al.

A comprehensive list of all undergraduate math courses at UTSA can be found here.

The Wikipedia summary of calculus and its history.

Topics List

Date Sections Topics Prerequisite Skills Student Learning Outcomes
Week 1
2.2

The Limit of a Function

  • Describe the limit of a function using correct notation.
  • Use a table of values to estimate the limit of a function or to identify when the limit does not exist.
  • Use a graph to estimate the limit of a function or to identify when the limit does not exist.
  • Define one-sided limits and provide examples.
  • Explain the relationship between one-sided and two-sided limits.
  • Describe an infinite limit using correct notation.
  • Define a vertical asymptote.


Week 1/2
2.3


The Limit Laws



  • Recognize the basic limit laws.
  • Use the limit laws to evaluate the limit of a function.
  • Evaluate the limit of a function by factoring.
  • Use the limit laws to evaluate the limit of a polynomial or rational function.
  • Evaluate the limit of a function by factoring or by using conjugates.
  • Evaluate the limit of a function by using the squeeze theorem.
  • Evaluate left, right, and two sided limits of piecewise defined functions.
  • Evaluate limits of the form K/0, K≠0.
  • Establish and use this to evaluate other limits involving trigonometric functions.
Week 2/3
2.4

Continuity


  • Continuity at a point.
  • Describe three kinds of discontinuities.
  • Define continuity on an interval.
  • State the theorem for limits of composite functions and use the theorem to evaluate limits.
  • Provide an example of the intermediate value theorem.


Week 3
4.6

Limits at infinity and asymptotes

  • Calculate the limit of a function that is unbounded.
  • Identify a horizontal asymptote for the graph of a function.



Week 3/4
3.1


Defining the Derivative

  • Recognize the meaning of the tangent to a curve at a point.
  • Calculate the slope of a secant line (average rate of change of a function over an interval).
  • Calculate the slope of a tangent line.
  • Find the equation of the line tangent to a curve at a point.
  • Identify the derivative as the limit of a difference quotient.
  • Calculate the derivative of a given function at a point.


Week 4
3.2


The Derivative as a Function

  • Define the derivative function of a given function.
  • Graph a derivative function from the graph of a given function.
  • State the connection between derivatives and continuity.
  • Describe three conditions for when a function does not have a derivative.
  • Explain the meaning of and compute a higher-order derivative.


Week 4/5
3.3


Differentiation Rules

  • State the constant, constant multiple, and power rules.
  • Apply the sum and difference rules to combine derivatives.
  • Use the product rule for finding the derivative of a product of functions.
  • Use the quotient rule for finding the derivative of a quotient of functions.
  • Extend the power rule to functions with negative exponents.
  • Combine the differentiation rules to find the derivative of a polynomial or rational function.


Week 5
3.4


Derivatives as Rates of Change

  • Determine a new value of a quantity from the old value and the amount of change.
  • Calculate the average rate of change and explain how it differs from the instantaneous rate of change.
  • Apply rates of change to displacement, velocity, and acceleration of an object moving along a straight line.
  • Predict the future population from the present value and the population growth rate.
  • Use derivatives to calculate marginal cost and revenue in a business situation.


Week 5
3.5


Derivatives of the Trigonometric Functions

  • Find the derivatives of the sine and cosine function.
  • Find the derivatives of the standard trigonometric functions.
  • Calculate the higher-order derivatives of the sine and cosine.


Week 6
3.6


The Chain Rule

  • State the chain rule for the composition of two functions.
  • Apply the chain rule together with the power rule.
  • Apply the chain rule and the product/quotient rules correctly in combination when both are necessary.
  • Recognize and apply the chain rule for a composition of three or more functions.
  • Use interchangeably the Newton and Leibniz Notation for the Chain Rule.


Week 6
3.7


Derivatives of Inverse Functions

  • State the Inverse Function Theorem for Derivatives.
  • Apply the Inverse Function Theorem to find the derivative of a function at a point given its inverse and a point on its graph.
  • Derivatives of the inverse trigonometric functions.


Week 6/7
3.8


Implicit Differentiation

  • Assuming, for example, y is implicitly a function of x, find the derivative of y with respect to x.
  • Assuming, for example, y is implicitly a function of x, and given an equation relating y to x, find the derivative of y with respect to x.
  • Find the equation of a line tangent to an implicitly defined curve at a point.


Week 7
3.9


Derivatives of Exponential and Logarithmic Functions

  • Know the properties associated with logarithmic expressions.
  • Rules for differentiating function (chain rule in particular).
  • Implicit differentiation.
  • Find the derivative of functions that involve exponential functions.
  • Find the derivative of functions that involve logarithmic functions.
  • Use logarithmic differentiation to find the derivative of functions containing combinations of powers, products, and quotients.


Week 7/8
4.1


Related Rates

  • Formulas from classical geometry for area, volume, etc.
  • Similar triangles to form proportions.
  • Right triangle trigonometry.
  • Use trigonometric identities to re-write expressions.
  • Rules for finding derivatives of functions.
  • Implicit differentiation.
  • Express changing quantities in terms of derivatives.
  • Find relationships among the derivatives in a given problem.
  • Use the chain rule to find the rate of change of one quantity that depends on the rate of change of other quantities.


Week 8
4.2


Linear Approximations and Differentials

  • Find the equation of the tangent line to a curve y = f(x) at a certain given x-value
  • Understand the Leibnitz notation of the derivative
  • Approximate the function value close to the center of the linear approximation using the linearization.
  • Given an expression to be evaluated/approximated, come up with the function and its linearization
  • Understand the formula for the differential; how it can be used to estimate the change in the dependent variable quantity, given the small change in the independent variable quantity.
  • Use the information above to estimate potential relative (and percentage) error


Week 8/9
4.3


Maxima and Minima

  • Understand the definition of an increasing and a decreasing function.
  • Solve an algebraic equation.
  • Understand interval notation.
  • Solve trigonometric equations.
  • Use all rules to differentiate algebraic and transcendental functions.
  • Understand definition of continuity of a function at a point and over an interval.
  • Know the definitions of absolute and local extrema.
  • Know what a critical point is and locate it (them).
  • Use the Extreme Value Theorem to find the absolute extrema of a continuous function on a closed interval.


Week 9
4.4


Mean Value Theorem

  • Function evaluation.
  • Solve equations.
  • Determine if the MVT applies given a function on an interval.
  • Find c in the conclusion of the MVT (if algebraically feasible)
  • Know the first 3 Corollaries of MVT (especially the 3rd)


Week 9
4.5


Derivatives and the Shape of a Graph

  • Function evaluation.
  • Solve equations.
  • Know how to find the derivative and critical point(s) of a function.
  • Know how to find the second derivative
  • Use the First Derivative Test to find intervals on which the function is increasing and decreasing and the local extrema and their type
  • Use the Concavity Test (aka the Second Derivative Test for Concavity) to find the intervals on which the function is concave up and down, and point(s) of inflection
  • Understand the shape of the graph, given the signs of the first and second derivatives


Week 10
4.7


Applied Optimization Problems

  • Translate the information given into mathematical statements/formulas.
  • Know frequently used formulas pertaining to area and volume.
  • Solve Algebraic and trigonometric equations.
  • Absolute extrema of a function


  • Set up a function to be optimized and find the value(s) of the independent variable which provide the optimal solution.


Week 10
4.8


L’Hôpital’s Rule

  • Simplifying algebraic and trigonometric expressions.
  • Evaluating limits.
  • Finding derivatives.
  • Identify indeterminate forms produced by quotients, products, subtractions, and powers, and apply L’Hôpital’s rule in each case.
  • Recognize when to apply L’Hôpital’s rule.


Week 11
4.10


Antiderivatives

  • Inverse Functions
  • Finding derivatives
  • Find the general antiderivative of a given function.
  • Explain the terms and notation used for an indefinite integral.
  • State the power rule for integrals.
  • Use antidifferentiation to solve simple initial-value problems.


Week 11/12
5.1


Approximating Areas

  • Sigma notation
  • Area of a rectangle
  • Graphs of continuous functions
  • Calculate sums and powers of integers.
  • Use the sum of rectangular areas to approximate the area under a curve.
  • Use Riemann sums to approximate area.


Week 12
5.2


The Definite Integral

  • Antiderivatives
  • Limits of Riemann Sums
  • Continuous functions over bounded intervals
  • State the definition of the definite integral.
  • Explain the terms integrand, limits of integration, and variable of integration.
  • Explain when a function is integrable.
  • Describe the relationship between the definite integral and net area.
  • Use geometry and the properties of definite integrals to evaluate them.
  • Calculate the average value of a function.


Week 12/13
5.3

The Fundamental Theorem of Calculus

  • Derivatives
  • Antiderivatives
  • Mean Value Theorem
  • Inverse functions
  • Describe the meaning of the Mean Value Theorem for Integrals.
  • State the meaning of the Fundamental Theorem of Calculus, Part 1.
  • Use the Fundamental Theorem of Calculus, Part 1, to evaluate derivatives of integrals.
  • State the meaning of the Fundamental Theorem of Calculus, Part 2.
  • Use the Fundamental Theorem of Calculus, Part 2, to evaluate definite integrals.
  • Explain the relationship between differentiation and integration.


Week 13
5.4


Integration Formulas and the Net Change Theorem

  • Indefinite integrals
  • Collections of functions
  • The Fundamental Theorem (part 2)
  • Displacment vs. distance traveled
  • Apply the basic integration formulas.
  • Explain the significance of the net change theorem.
  • Use the net change theorem to solve applied problems.
  • Apply the integrals of odd and even functions.



Week 14
5.5


Substitution Method for Integrals

  • Solving basic integrals.
  • Derivatives
  • Change of Variables
  • Use substitution to evaluate indefinite integrals.
  • Use substitution to evaluate definite integrals.



Week 14/15
5.6


Integrals Involving Exponential and Logarithmic Functions

  • Exponential and logarithmic functions
  • Derivatives and integrals of these two functions
  • Rules for derivatives and integration
  • Integrate functions involving exponential functions.
  • Integrate functions involving logarithmic functions.


Week 15
5.7


Integrals Resulting in Inverse Trigonometric Functions

  • Trigonometric functions and their inverses
  • Injective functions and the domain of inverse trigonometric functions
  • Rules for integration
  • Integrate functions resulting in inverse trigonometric functions.