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B1.5 – Derivatives of Secondary Trig Functions

B1.5 – Derivatives of Secondary Trig Functions. IB Math HL - Santowski. For y = sec(x) on (-2 ,2) Fcn is con up on (-2 ,-3/2), (-/2, /2), (3/2,2) Fcn is con down elsewhere Fcn has max at -,  Fcn has min at -2,0,2 Fcn increases on (-2,-3/2), (-3/2,-), (0,/2), (/2,)

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B1.5 – Derivatives of Secondary Trig Functions

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  1. B1.5 – Derivatives of Secondary Trig Functions IB Math HL - Santowski

  2. For y = sec(x) on (-2,2) Fcn is con up on (-2,-3/2), (-/2, /2), (3/2,2) Fcn is con down elsewhere Fcn has max at -,  Fcn has min at -2,0,2 Fcn increases on (-2,-3/2), (-3/2,-), (0,/2), (/2,) Fcn decreases elsewhere Fcn has asymptotes at +3/2, +/2 (A) Derivatives of f(x) = sec(x) - Graphically

  3. For y = sec(x) on (-2,2) Fcn is con up on (-2,-3/2), (-/2, /2), (3/2,2)   f ` increases here Fcn is con down elsewhere   f ` decreases here Fcn has max at -,   roots on f ` Fcn has min at -2,0,2  roots on f ` Fcn increases on (-2,-3/2), (-3/2,-), (0,/2), (/2,)  f ` is positive Fcn decreases elsewhere  f ` is negative Fcn has asymptotes at +3/2, +/2  f ` has asymptotes (A) Derivatives of f(x) = sec(x) - Graphically

  4. (B) Derivative of f(x) = sec(x) - Algebraically • We will use the fact that sec(x) = 1/cos(x) to find the derivative of sec(x)

  5. (C) Derivatives of f(x) = csc(x) and f(x) = cot(x) • We can run through a similar curve analysis and derivative calculations to find the derivatives of the cosecant and cotangent functions as well • The derivatives turn out to be as follows: • d/dx csc(x) = -csc(x) cot(x) • d/dx cot(x) = -csc2(x)

  6. primary trig fcns: secondary trig fcns: (D) Summary of Trig Derivatives

  7. (E) Examples • (i) Differentiate • (ii) Differentiate • (iii) find dy/dx if tan(y) = x2 • (iv) find the slope of the tangent line to y = tan(csc(x)) when sin(x) = 1/ on the interval (0,/2) • (v) Find the intervals of concavity of y = sec(x) + tan(x)

  8. (F) Homework • Stewart, 1989, Chap 7.3, p319, Q1-4,6,7

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