MATH 118 – Fundamental Principles of the Calculus

Math 118 is a coordinated course and a common final exam is offered at the end of each semester.
You can find past common finals here.

REQUIRED TEXTBOOK

Hughes-Hallet/Gleason/Lock.Flath/et.al., Applied Calculus (7th ed.)

Math 118 uses the accompanying WileyPLUS homework system which is integrated into Blackboard.  Clicking any of the homework sets in Blackboard should prompt students to purchase the homework and textbook package.  The USC bookstore also carries physical copies of the book with an activation code for WileyPLUS.

SECTION COVERAGE

The following table lists the minimum set of topics to be included in this course.  The number of lectures listed for each chapter is only a suggestion and will vary across instructors and semesters.

Math 118 specifically omits all trigonometry and trigonometric functions, notably Sections 1.10 and 3.5.

There are typically 42-43 lecture days in a semester, so lecture periods are available for exams.

OPTIONAL TOPICS (time permitting)

• Sections 4.7 and 4.8, different applications of derivatives
• Section 6.7 on integration by parts
• Other outside topics, readings, videos, or materials the Instructor deems relevant

If all course instructors in a given semester cover a particular topic, they may choose to include that topic in the common final exam.

MEASURABLE OBJECTIVES

Linear Functions and Average Rates of Change:

• Students can recognize a linear function from a graph or table of values and find a formula for the function.
• Students can identify and correctly interpret the slope and intercept of a linear function in context and identify their units.
• Students know the definitions of increasing/decreasing and concave up/concave down, can interpret them in context, can identify intervals where a function has these properties given a graph or table of values, and can relate the latter to the non-linearity of the function.
• Students can compute, compare, and interpret average rates of change of a function from formulae, graphs, and tables of values with attention to units.
• Students can use average rates of change to estimate and compute absolut changes in the function and can identify if the estimates are over-estimates or under-estimates.

Exponential Functions:

• Students can compute and compare absolute and relative changes and, given a relative change, can compute the value of the function.
• Students can recognize an exponential function by how the outputs change when the inputs change and can find an exponential function that fits two data points.
• Students can find when an exponential function takes on a particular value and can find the time taken to increase/decrease by a certain factor.
• Students can reconcile and interpret the three different representations of an exponential function.

Instantaneous Rates of Change:

• Students can define the instantaneous rate of change or derivative of a function and relate it to the slope of the graph.
• From a graph or table of values of a function, students can identify locations where the derivative is positive, negative, increasing and decreasing, and can sketch a graph of the derivative.
• Students can interpret and use Leibniz and function notation for a derivative and can use units to interpret the meaning of a derivative in context.
• Students can estimate the derivative given a formula, graph, or table of values of a function.
• Students can use the derivative to estimate values of the function and identify if the estimates are over-estimates or under-estimates.
• Students can compute and correctly interpret relative rates of change.
• Students can define the second order derivative, relate it to the rate of change of the derivative and to the concavity of the function and estimate its value given a formula or table of values of a function with attention to units.

Techniques of Differentiation:

• Students can find a formula for the derivative of a function when the function is given as a formula, using the product, quotient, and chain rules as appropriate.
• Students can use the product, quotient, and chain rules to find the derivatives of functions created from other functions.

Optimization, and Graphical Properties of a Function and Its Derivatives:

• Students can identify the critical points, local extrema, global extrema, and inflection points of a function from a graph of the function.
• Students can identify the locations of critical points, local extrema, global extrema, and inflection points of a function from a graph of its derivative.
• Students can identify the shape of the graph of the function at different points from the graph of its derivative.
• Students can find the critical points of a function given as a formula and use the first or second derivative test to classify each as a local max, local min, or neither.
• Students can determine if a function has global extrema on an interval and find them when they exist.

Applications of Derivatives to Business and Economics:

• Students can use functions to model and represent economic concepts such as cost, revenue, profit, supply, and demand and can recognize and distinguish the fixed costs and variable costs given a cost function.
• Students can define marginal cost, revenue and profit, can interpret them in context, and can relate them to and distinguish them from cost, revenue and profit.
• Students can define average cost per item and can visualize it and compare it with marginal cost given a graph of the cost function.
• Students can construct and use functions to make optimal decisions (such as maximizing profit, revenue, and efficiency and minimizing cost).
• Students can define elasticity of demand, interpret it in context, estimate an compute its value, use it to relate relative changes in price with relative changes in quantity, and relate it to the revenue function.

Definite Integrals, Antiderivatives, and the Fundamental Theorem of Calculus:

• Students can define a definite integral as a limit of Riemann sums. Students can estimate the value of a definite integral using a finite Riemann sum and by computing areas under the graph. Students can find upper or lower estimates if desired and can compute a bound on the error when the function is monotonic.
• Given a rate of change function as a formula, graph, or table of values, students can correctly interpret, estimate, and compute total change as a definite integral.
• Students know the definition of an antiderivative and know how to check if a function is an antiderivative of another function.
• Students can identify the family of antiderivatives from a specific antiderivative and can find a specific antiderivative with a specified value given the family of antiderivatives.
• Students can state the Fundamental Theorem of Calculus in multiple ways.
• Students can use the Fundamental Theorem to graph a function given the graph of its derivative, to identify properties of antiderivatives given the function graphically, and to find the specific antiderivative with a specified value.

Techniques of Integration:

• Students can find the family of antiderivatives of a function that is given by a formula using basic rules and substitutions.
• Students can use the Fundamental Theorem of Calculus and techniques of integration to evaluate definite integrals.

Applications of Definite Integrals to Business and Economics:

• Students know the meaning and definition of the average value of a function, can compute its value, and can recognize it in context.
• Students can compute and estimate total and average cost, profit, and revenue given marginal cost, profit, and revenue. Students can distinguish between average cost and the average value of the marginal cost.
• Students can define and interpret consumer and producer surplus, relate them to areas under the supply and demand curves, and compute their values given the supply and demand curves.
• Students can interpret the present, future, or total value of an income stream in context and compute their values.

Multivariable Functions:

• Students can correctly use function notation in the context of functions of more than one variable. Students can define and interpret level sets and cross-sections of a function of more than one variable. Students can identify properties of functions of two variables from formulae, tables of values, and contour diagrams.
• Students can define the partial derivatives of a function of more than one variable, can interpret a partial derivative in context, and can use Leibniz and function notation to notate them.
• Students can estimate and compute partial derivatives from formulae, tables of values and contour graphs. Students can recognize from a table of values or contour graph locations where a partial derivative is equal to zero.
• Students can use partial derivatives to estimate values of the function.

Optimization of Functions of Two Variables:

• Students can identify local extrema of functions of two variables from contour graphs and tables of values.
• Students can find local extrema of functions of two variables and classify them as local max, min, or saddle points.
• Students can identify the global extrema of a function of two variables subject to a constraint from a contour graph of the function.
• Students can use the method of Lagrange multipliers to find the global extrema of a function of two variables subject to a constraint and interpret the value of lambda in context.