###### Learning Outcomes:

On completion of this module students should be able to:1. Calculate limits.2. Calculate derivatives via implicit and logarithmic differentiation. 3. Display knowledge of the properties of polynomial and rational functions, trigonometric functions and their inverses, exponential, logarithmic and hyperbolic functions. 4. Use calculus to find local extrema of a function of one variable and apply these methods to optimisation problems. 5. Calculate definite and indefinite integrals. 6. Determine the Interval the Convergence of a power series. Obtain the expansion for MacLaurin and Taylor series of a function of a single variable. 7. Solve first order linear differential equations .

###### Indicative Module Content:

Functions and Limit

Definition of a function; bijective and inverse functions

Geometric criteria for bijective and inverse functions

Limit of a function. Computing the limit by multiplying with the rational conjugate

Continuous functions

Differentiation

Definition. Mechanical and geometrical interpretation of derivati

Basic rules of differentiation (sum, difference, product, quotient, chain rule)

Logarithmic differentiation

Implicit differentiation; the eq. of the tangent line to a curve

Trigonometric functions $\sin x$ and $\cos x$

]The inverse trig. functions $\sin^{-1} x$ and $\cos^{-1} x$ and their derivatives

Hyperbolic functions

The hyperbolic functions

Inverse hyperbolic functions their graphs and derivatives

Various identities for inverse hyperbolic functions

Optimisation

Critical points of functions of one variable

Maximum, minimum and point of inflection

Convexity and concavity

The Second Derivative Test for functions of one variable.

Integral Calculus-Methods of integration

Integration by parts, LIATE rule

Integration by substitution

Method of partial fractions

mproper integrals

Numerical Approximation of integrals

Midpoint rule, Trapezoidal rule and Simpson's rule

Applications of Integral Calculus} (Formulae provided on question sheet)

Total area

Area of the region between two graphs

Volume of a solid of rotation

Length of a Graph

Area of a Surface:

Differential equations

First order differential equations with separable variables

Applications from Physics and Chemistry

Linear first order differential equations . General solution using Integrating factor

Sequences and series

Geometric series

Harmonic series test

Ratio test

Alternating series test and Comparison series test

Power series

Interval of convergence, radius of convergence for a power series

Taylor series

Maclaurin series

Applications of Differential Equations

Applications of Differential Equations using Separation of Variables

Applications of Differential Equations using Integrating Factor