MAT1214

Topics List

Topic	Pre-requisite	Objective	Examples
The Limit of a Function	Evaluation of a function including the absolute value, rational, and piecewise functions Domain and Range 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.	Edit soon
The Limit Laws	Simplifying algebraic expressions. Factoring polynomials Identifying conjugate radical expressions. Evaluating expressions at a value. Simplifying complex rational expressions by obtaining common denominators. Evaluating piecewise functions. The trigonometric functions and right triangle trigonometry.	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.	edit soon
Continuity	Domain of function. Interval notation. Evaluate limits.	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.	Edit
Limits at infinity and asymptotes	Horizontal asymptote for the graph of a function	Calculate the limit of a function that is unbounded. Identify a horizontal asymptote for the graph of a function.	Edit
Defining the Derivative	Evaluation of a function at a value or variable expression. Find equation of a line given a point on the line and its slope. Evaluate limits.	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.	Edit
The Derivative as a Function	Graphing functions. The definition of continuity of a function at a point. Understanding that derivative of a function at a point represents the slope of the curve at a point. Understanding when a limit fails to exist.	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.	Edit
Differentiation Rules	Radical and exponential notation. Convert between radical and rational exponents. Use properties of exponents to re-write with or without negative exponents.	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.	Edit
Derivatives as Rates of Change	Function evaluation at a value or variable expression. Solving an algebraic equation. Find derivatives of functions using the derivative rules.	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.	Edit
Derivatives of the Trigonometric Functions	State and use trigonometric identities. Graphs of the six trigonometric functions. Power, Product, and Quotient Rules for finding derivatives.	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.	Edit
The Chain Rule	Composition of functions. Solve trigonometric equations. Power, Product, and Quotient Rules for finding derivatives.	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.	Edit
Derivatives of Inverse Functions	Determine if a function is 1-1. The relationship between a 1-1 function and its inverse. Knowing customary domain restrictions for trigonometric functions to define their inverses. Rules for differentiating 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.	Edit
Implicit Differentiation	Implicit and explicit equations. Point-slope and slope-intercept equation of a line. Function evaluation. Know all rules for differentiating functions.	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.	Edit
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.	Edit
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.	Edit
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	Edit
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.	Edit
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)	Edit
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	Edit
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.	Edit
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.	Edit
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.	Edit
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.	Edit