ENGI 345 Review of Calculus Page 1.01 1. Review of Calculus We begin this course with a refresher on ifferentiation an integration from MATH 1000 an MATH 1001. 1.1 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
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
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 1.1.1 Show that a u x u x u x a ln a x
ENGI 345 Review of Calculus Page 1.04 1. 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 1..3 1 xln x x ux u x e x Example 1..4 x3 x e x
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
ENGI 345 Review of Calculus Page 1.06 a 1 x x a 1 x x a 1 x x a 1 x x
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.
ENGI 345 Review of Calculus Page 1.08 1.3 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.
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.
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 1 1 + tan sec
ENGI 345 Review of Calculus Page 1.11 Derivatives x sinh x x cosh x x tanh x
ENGI 345 Review of Calculus Page 1.1 x csch x En of Chapter 1