The primary aim of MTH 141 is to prepare students for further study in mathematics, basic sciences, or engineering by providing an introduction to differential and integral calculus, and by helping students develop new problem solving and critical reasoning skills. The objectives of MTH 141 are

1. To provide a thorough introduction to differential calculus concepts and methods.
2. To provide an introduction to integration as a limit of sums, and to the Fundamental Theorem of calculus.
3. To provide an introduction to mathematical modeling and numerical issues through the use of technology.

Computation of limits by graphical, numerical, and algebraic methods, and the use limits and theorems on continuity to determine continuity properties of functions.

Computation of derivatives using difference quotients by graphical, numerical and algebraic methods, interpretation of derivatives as rates of change and as slopes of tangent lines, and the use theorems on differentiation (both for computation of derivatives or for properties of differentiable functions).

Determination of critical and inflection points of functions by graphical and algebraic methods, the use first or second derivatives to analyze monotonicity and concavity of functions, determination of local and global optima of functions.

Application of derivatives to the computation of limits (L’Hopital’s rule), computation of derivatives of functions defined implicitly.

Computation of Left-Sums and Right-Sums of functions given algebraically, in tabular form or graphically, and their use to approximate areas and integrals. Interpretation of integrals of rates of change as total change, the use theorems on integration to compute simple integrals, and to determine properties of functions given algebraically or graphically.

At the end of the course the student should be able to:

Evaluate the limit of a function at a point or at infinity analytically, graphically, and numerically.

Evaluate two-sided limits analytically, graphically, and numerically.

Compute limits that result in infinity, and use this to support statements about the nature of the function.

Use limits to determine vertical and horizontal asymptotes of a function

Use limits to determine if a function is continuous at a point.

For a function given in algebraic or graphical form and defined on an interval or union of intervals, establish if it is continuous in its domain.

Use theorems on continuity of addition, product, quotient and composition of continuous functions to determine if a function is continuous at a point or on an interval.

For a given function, calculate the average rate of change over a given interval.

For a given function, approximate the instantaneous rate of change over a given interval.

Approximate numerically or graphically the slope of a tangent line to a curve at a point.

Define and evaluate the derivative at a point as a limit.

Compute algebraically the derivative function using limits.

Approximate numerically or graphically the derivative of a function at a point.

Given the plot of a function, plot the derivative function.

Use graphical, numerical, or algebraic arguments to study differentiability at a point.

Recall and use the derivative of the functions: constant, power, logarithmic, exponential, trigonometric, inverse trigonometric, hyperbolic.

Use theorems of derivatives: linearity, product rule, quotient rule, chain rule.

Compute the derivative of a function given implicitly, and determine slopes to curves defined implicitly.

Compute higher order derivatives.

Use derivatives to compute velocity and acceleration of bodies when the displacement function is given.

Use derivatives to solve related rates problems.

Determine critical points and inflection points of a function given algebraically or graphically.

Use derivatives to determine intervals where a given function is increasing or decreasing, and where a function is concave up or concave down.

Find local optima by finding critical points and then using the first or second derivative tests.

Determine global optima of a function defined on a bounded or unbounded interval.

Use derivatives to determine intervals where a function is increasing, decreasing, concave up, concave down.

Find the linear approximation to a function at a given point.

Use L'Hopital's rule to compute limits of the indeterminate forms "zero over zero" and "infinity over infinity".

Use derivatives to compute slopes and tangent lines to curves in the plane given parametrically.

Compute the derivative of functions given in terms of one or more parameters, and interprete derivatives and the function in relation to parameter values.

State theorems about continuous and differentiable functions, and be able to use them in simple, direct applications. (Extreme Value Theorem, Mean Value Theorem, Rolle's theorem, the Racetrack Principle.)

Write down Riemann sums for functions given algebraically.

Represent Riemann sums graphically (left sum, right sum, other).

Use Riemann sums to obtain approximations to areas under a curve or to a definite integral.

Given the graph of a function, approximate the value of an integral by using Riemann sums.

Given a table of values, obtain a Riemann sum and approximate a definite integral.

Compute areas between curves using integrals.

Interpret total change in a function as an integral of rate of change.

State and use the (first form of the ) Fundamental Theorem of Calculus to compute integrals.

State and use the (Second form of the ) Fundamental Theorem of Calculus to compute integrals.

Use linearity and additive properties to compute integrals.

Express, in natural language, model characteristics that are given mathematically through equations or graphs.

Develop a mathematical model from natural language specification, graphs, geometric figures.

Reason symbolically with parameters to determine properties of a model.

Recognize applicability of a model to a situation and limitations imposed by assumptions.

Recognize patterns, trends, symmetries, and use these to formulate conjectures or draw conclusions

Formulate valid arguments to support or refute a conjecture or hypothesis

Determine validity in an argument or identify the flaw in an invalid argument.

Use problem solving strategies such as considering particular cases, drawing figures, establishing similarities with other problems, etc.

Determine if an approximation is accurate to a number of digits.

Recognize reasonableness of a result through the use of approximations or by checking order of magnitude, correct units, appropriate signs, etc.