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