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Trig Identities and Applications

This lesson focuses on integrating trigonometric functions and understanding their derivatives, such as sin(x), cos(x), and tan(x). Students will learn how to apply these functions in real-world problems involving position, velocity, and acceleration. Key topics include writing position functions and solving differential equations to find unknown constants. Through practical examples, students will resolve various integration challenges and gain insights into the applications of antiderivatives in physics, particularly in motion problems, including projectile motion and height calculations.

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Trig Identities and Applications

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  1. Trig Identities and Applications A calc student upset as could be That his antiderivative didn't agree With the one in the book E'en aft one more look. Oh! Seems he forgot to write the "+ C".

  2. Objective • To integrate trig functions • To write position, velocity, and acceleration functions

  3. 4.1 Trig Identities and applications • What are the derivatives of… • sin x • cos x • tan x • sec x • csc x • cot x

  4. Trig identities

  5. Examples

  6. Words that mean integration… • Anti-derivative • Integral • General solution of a differential equation • Area under the curve • Position function

  7. Applications… solving for C • Given: f’(x)=2, f(3)=5 • Find: f(x)

  8. You try… • Given: f’(x) = x^2, f(2)=0. • Find f(x) • Given: f’’(x) = x, f’(0)=4, f(0)=2. • Find f(x)

  9. Position, velocity, and acceleration funcitons s(t) v(t) a(t)

  10. Ex 8, pg 247 • A ball is thrown upward with an initial velocity of 64 ft/sec from an initial height of 80 feet. • a) What is the position function?

  11. Ex 3…. • A ball is thrown upward with an initial velocity of 64 ft/sec from an initial height of 80 feet. • b) When does the ball hit the ground?

  12. Problem 63, pg 250 • A baseball is thrown upward from ground level with a velocity of 10 m/sec. What is it’s maximum height?

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