Theory questions

One of the problems on part C the exam will be a theoretical one. You might be asked to write down or use a definition, to formulate and prove a theorem or something like that. You might also be asked to give or analyse specific examples. The following might be part of such a problem on the exam:

• Definition of limit (ch. 1.5, definition 8 and definition 10)
• Definition of continuity (ch. 1.4, definition 4)
• Definition of the derivative (ch. 2.2, definition 4)
• Differentiable functions are continuous (ch. 2.3, theorem 1 and its proof)
• Product rule and quotient rule  (ch. 2.3, theorems 3 and 5 and their proofs)
• The chain rule (ch. 2.4, theorem 6 and its proof)
• The mean value theorem and its consequences (ch. 2.8, theorems 11, 12, 13, 14 and 15 and proofs)
• Existence och localisation of  extreme values (ch. 4.4, theorem 5 (without proof) and theorem 6 (with proof)
• Definition of linearization (ch. 4.9, definition 8)
• Taylor's formula (ch 4.10, theorem 12 without proof)
• The mean value theorem for integrals  (ch. 5.4, theorem 4 with proof)
• The fundamental theorem of calculus (ch. 5.5, theorem 5 with proof)
• Substitution in integrals (ch. 5.6, theorem 6 with proof)
• Integration by parts (ch. 6.1, the formula on the bottom of page 332 and its derivation)
• Convergence of a sequence of numbers  (ch. 9.1, definition 2)
• Convergence of a series  (ch. 9.2, definition 3)
• Integral criterion for series (ch. 9.3, theorem 8)
• Comparison test for series (ch. 9.3, theorem 9)