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B.1.8 – Derivatives of Primary Trig Functions

B.1.8 – Derivatives of Primary Trig Functions. Calculus - Santowski. Background Skills. 1. Graphs of sin(x), cos(x), tan(x) & their features 2. Evaluating trig functions at a given point (No calc) 3. Trig identities – Pythagorean, add/sub, double angle 4. Solving simple trig equations.

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B.1.8 – Derivatives of Primary Trig Functions

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  1. B.1.8 – Derivatives of Primary Trig Functions Calculus - Santowski Calculus - Santowski

  2. Background Skills • 1. Graphs of sin(x), cos(x), tan(x) & their features • 2. Evaluating trig functions at a given point (No calc) • 3. Trig identities – Pythagorean, add/sub, double angle • 4. Solving simple trig equations Calculus - Santowski

  3. Fast Five • 1. State the value of sin(/4), tan(/6), cos(/3), sin(/2), cos(3/2) • 2. Solve the equation sin(2x) - 1 = 0 • 3. Expand sin(x + h) • 4. State the value of sin-1(0.5), cos-1(√3/2) • 5. Explain how to find the value of the limx->0 (cosx - 1)/x • 6. Explain how to find the value of the limx->0 sin(x)/x Calculus - Santowski

  4. Lesson Objectives • 1. Review the derivatives of trigonometric functions algebraically and graphically • 2. Differentiate equations involving trigonometric functions • 3. Apply sinusoidal functions and their derivatives to questions involving tangent lines and function analysis Calculus - Santowski

  5. (A) Review - Derivatives • The derivative of y = sin(x) is dy/dx = cos(x) Calculus - Santowski

  6. Trig Derivatives • Using the derivative of y = sin(x) and your fundamental basic trig knowledge, determine the derivatives of the other 5 trig functions • (a) y = cos(x) • (b) y = tan(x) • (c) y = sec(x) • (d) y = csc(x) • (e) y = cot(x) Calculus - Santowski

  7. Differentiating with sin(x) & cos(x) • Differentiate the following • y = cos(x2) • y = cos2(x) • y = 3sin(2x) • y = 6xsin(3x2) • Differentiate the following: Calculus - Santowski

  8. Applications – Tangent Lines • Find the equation of the tangent line to f(x) = xsin(2x) at the point x = π/4 • What angle does the tangent line to the curve y = f(x) at the origin make with the x-axis if y is given by the equation Calculus - Santowski

  9. Applications – Curve Analysis • Find the maximum and minimum point(s) of the function f(x) = 2cosx + x on the interval (-π,π) • Find the minimum and maximum point(s) of the function f(x) = xsinx + cosx on the interval (-π/4,π) • Find the interval in which g(x) = sin(x) + cos(x) is increasing on xER Calculus - Santowski

  10. Applications • Given • (a) for what values of a and b is g(x) differentiable at 2π/3 • (b) using the values you found for a & b, sketch the graph of g(x) Calculus - Santowski

  11. Internet Links • Calculus I (Math 2413) - Derivatives - Derivatives of Trig Functions from Paul Dawkins • Visual Calculus - Derivative of Trigonometric Functions from UTK • Differentiation of Trigonometry Functions - Online Questions and Solutions from UC Davis • The Derivative of the Sine from IEC - Applet Calculus - Santowski

  12. Homework • Handout from Stewart, Calculus: A First Course, 1989, Chap 7.2, Q1&3 as needed, 4-7,9 Calculus - Santowski

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