ENGI 3425 Review of Calculus Page then

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1 ENGI 345 Review of Calculus Page Review of Calculus We begin this course with a refresher on ifferentiation an integration from MATH 1000 an MATH Reminer of some Derivatives (review from MATH 1000) Prouct Rule: x uv Quotient Rule: x u v Chain Rule: If y f u an u g x then x n x e kx x x ln x x sin x x cos x

2 ENGI 345 Review of Calculus Page 1.0 x tan x x csc x x sec x x cot x ux e x x u x n x sin n ux x cos n ux ux a x

3 ENGI 345 Review of Calculus Page 1.03 x ln u x Algebra of exponents: e u v u e v e Algebra of logarithms: ln uv ln u v n ln x Double angle formulae: cos sin sin cos Implicit Differentiation Example Show that a u x u x u x a ln a x

4 ENGI 345 Review of Calculus Page Reminer of some Integrals (review from MATH 1001) n u x u x x Example 1..1 tan 4 xsec x x u x x u x Example 1.. tan x x Example xln x x ux u x e x Example 1..4 x3 x e x

5 ENGI 345 Review of Calculus Page 1.05 cos u x u x x Example 1..5 x x e cose x Some trigonometric integrals: sin cos sec x x sec x tan x x sec x x csc x x csc xcot x x csc x x

6 ENGI 345 Review of Calculus Page 1.06 a 1 x x a 1 x x a 1 x x a 1 x x

7 ENGI 345 Review of Calculus Page 1.07 Integration by parts (tabular version) the table terminates in one of three ways: 1. Left column ens in zero (one factor in the integran must be a polynomial). Example 1..6 x 1 cos x x. The last row is easily integrate: Example 1..7 x n ln x x 3. The last row is a constant multiple of the original integran: Example 1..8 ax I e sin bx x ENGI 345 assumes mastery of the concepts an techniques in MATH 1000, 1001 an 050.

8 ENGI 345 Review of Calculus Page Hyperbolic Functions When a uniform inelastic (unstretchable) perfectly flexible cable is suspene between two fixe points, it will hang, uner its own weight, in the shape of a catenary curve. The equation of the stanar catenary curve is most concisely expresse as the hyperbolic cosine function, y cosh x, where e e cosh x The solutions to some ifferential equations can be expresse conveniently in terms of hyperbolic functions. Another hyperbolic function is x x e e sinh x The graphs of these two hyperbolic functions are isplaye here: x x The other four hyperbolic functions are x x sinh x e e 1 tanh x x x, sech x x x, cosh x e e cosh x e e x x 1 e e 1 coth x x x an csch x x x. tanh x e e sinh x e e Unlike the trigonometric functions, the hyperbolic functions are not perioic. However, parity is preserve: Of the six trigonometric function, only cos an sec are even functions. Of the six hyperbolic functions, only cosh x an sech x are even functions. The other four trigonometric functions an the other four hyperbolic functions are all o.

9 ENGI 345 Review of Calculus Page 1.09 Graphs of the other four hyperbolic functions: The solutions to some orinary ifferential equations that moel logistic growth (constraine growth) of a population resemble the hyperbolic tangent function.

10 ENGI 345 Review of Calculus Page 1.10 There is a close relationship between the hyperbolic an trigonometric functions. From the Euler form for e j, e j = cos + j sin, (where j 1 ) Ientities: Let x = j : sin + cos tan sec

11 ENGI 345 Review of Calculus Page 1.11 Derivatives x sinh x x cosh x x tanh x

12 ENGI 345 Review of Calculus Page 1.1 x csch x En of Chapter 1