# Further Calculus – LSE

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This course follows on from Calculus and Algebra, and continues further the study of calculus techniques and theory. The course will develop further the theory of functions, and will also include some new practical skills, such as how to evaluate double integrals and how to use Laplace transforms to solve differential equations.

• Functions of one variable: Limits; continuity; differentiability; Taylor’s Theorem; L’Hôpital’s rule.
• The Riemann integral: The definition of the Riemann integral; the Fundamental Theorem of Calculus.
• Improper integrals: The definition of an improper integral; tests for the convergence of an improper integral with a positive integrand (including the direct comparison test and the limit comparison test); absolute convergence of improper integrals with an integrand of variable sign.
• Double integrals: Double integrals; repeated integrals; change of variable techniques.
• Manipulation of integrals: Joint continuity and the manipulation of proper integrals; dominated convergence and the manipulation of improper integrals; the Leibniz rule for differentiating an integral.
• Laplace transforms: The definition of the Laplace transform; functions of at most exponential growth; standard Laplace transforms; properties of the Laplace transform; the Gamma function; using Laplace transforms to solve differential equations; convolutions and the Convolution Theorem; the Beta function.

### Learning outcomes

If you complete the course successfully, you should be able to:

• Demonstrate knowledge of the subject matter, terminology, techniques and conventions covered in the subject
• Demonstrate an understanding of the underlying principles of the subject
• Demonstrate the ability to solve problems involving an understanding of the concepts