Table of Contents 13. Section 3.5 and 3.6 Higher Derivatives and Trig Functions

Higher Derivatives and Trig Functions Essential Question – How do you find derivatives for trig functions?

Second and Higher Order Derivatives y’ is called 1 st derivative y’’ is called 2 nd derivative y’’’ is called 3 rd derivative y (4) is called the 4 th derivative

Example Find 1 st 4 derivatives of and evaluate y’’’(-1)

Example Calculate f’’’(x) for f(x) = xe x

Graphing calculator You can graph a derivative function without calculating it by using the nDeriv function (Math 8) Put nDeriv(f(x), x, x) into y= f(x) is the function, the first x means you are taking derivative with respect to x, the second x means that you want for all values of x (you could put a number here and it would just calculate that one value)

Sin and Cos Graph Nderiv(sin x, x, x) Can you tell what the derivative of sin is? y’(sin x) = cos x Graph Nderiv(cos x, x, x) Can you tell what the derivative of sin is? y’(cos x) = - sin x

Example Find derivative

Example

Simple Harmonic Motion A weighted spring is an example of simple harmonic motion Its motion is modeled by cos.

Example A weight hanging from a spring is stretched 5 units past its rest position (s=0) and released at time 0 (t=0) to bob up and down. Its position at any later time is Find the velocity and acceleration at time t.

Other trig function rules

Example Verify the formula

Example Find the tangent line to the function below at x=2.

Example Find y” if y = sec x

Assignment Pg. 165: # 1, 5, 9, 13, 17 Pg. 170: # 1, 5, 9, 13, 17, 21, 25, 29, 33, all