# MA1113 Single Variable Calculus I

Review of analytic geometry and trigonometry, functions of one variable, limits, derivatives, continuity and differentiability; differentiation of algebraic, trigonometric, logarithmic and exponential functions with applications to maxima and minima, rates, differentials; product rule, quotient rule, chain rule; anti-derivatives, integrals and the fundamental theorem of calculus; definite integrals, areas. Taught at the rate of nine hours per week for five weeks. Prerequisites: None.

### Lecture Hours

4### Lab Hours

0### Course Learning Outcomes

Functions and limits

· Use real numbers, inequalities involving real numbers and their absolute values, the trigonometric functions, and the radian measure of angles.

· Be able to move back and forth between the descriptions of a function by an equation, a table, a graph, and by words.

· Be able to use exponential functions, sketch their graphs, and define the number e.

· Define what it means for a function to be one-to-one and determine whether a function has an inverse or not and sketch its inverse if it does.

· Be able to use logarithmic functions, sketch their graphs, and define the relationship between the natural exponential and natural logarithmic functions.

· State in words what it means for a function to have a limit, be able to calculate limits, and be able to find the vertical and horizontal asymptotes of a function.

· State in words what it means for a function to be continuous and be able to find limits for continuous functions.

Derivatives

· Relate the notions of tangent to a curve, velocity, and rate of change, and illustrate them in a sketch.

· State the definition of derivative as the limit of a difference quotient and explain how the derivative itself can be regarded as a function.

· Be able to find derivatives of polynomials and exponential functions.

· State the product and quotient rules for differentiation and be able to use them to differentiate functions.

· Know the derivatives of sine and cosine and be able to use the quotient rule to determine the derivatives of the remaining four trigonometric functions.

· State the chain rule and use it to differentiate functions obtained by composition.

· Use the differentiation rules to differentiate implicitly, and to find higher order derivatives.

· Be able to differentiate logarithmic functions, and functions involving them.

· Define the hyperbolic functions and be able to differentiate them.

Applications of Differentiation

· Be able to solve related rates problems. Understand them as an application of the chain rule.

· Understand the connection between the derivative, the tangent line to the graph of a function, the linearization of a function, and the differential of a function.

· Use the differential (or linearization) to solve “small change” and applied approximation problems.

· Be able to state the Mean Value Theorem and give some of its consequences.

· Describe how the signs of the first and second derivatives of a function affect the shape of its graph.

· Define and recognize the various forms of indeterminate forms, and use L”Hospital’s Rule to determine their limits.

· Be able to set up and solve optimization problems using calculus methods.

· Be able to describe Newton’s method geometrically, and to use it to iteratively approximate the zeros of functions.

· Define what the antiderivative of a function is and be able to find it for reasonable functions.

Integral Calculus

· Describe the connection between the problems of finding areas and distances travelled, and how both problems lead to the same limit.

· Know and be able to work with the properties of definite integrals.

· State the Fundamental Theorem of Calculus in words, describe how it connects integral and differential calculus, and how it helps in finding antiderivatives and in evaluating definite integrals.

· Define the indefinite integral of a function and state its relation to the antiderivative.

· Be able to use the Substitution Rule to evaluate definite and indefinite integrals.

· Be able to use integration by parts to evaluate appropriate integrals.