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1. Find the derivative 2. Draw the graph of y’ from the graph of f(x)
Warm-up 1. Find the derivative 2. Draw the graph of y’ from the graph of f(x)
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12. Section 3.5 and 3.6 Higher Derivatives and Trig Functions
Table of Contents 12. Section 3.5 and 3.6 Higher Derivatives and Trig Functions
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Higher Derivatives and Trig Functions
Essential Question – How do you find derivatives for trig functions?
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Second and Higher Order Derivatives
y’ is called 1st derivative y’’ is called 2nd derivative y’’’ is called 3rd derivative
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Example Find 1st 4 derivatives of and evaluate y’’’(-1)
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Example Calculate f’’’(x) for f(x) = xex
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Acceleration Acceleration is a common second derivative
Find the acceleration of a ball tossed vertically in the air from ground level with an initial velocity of 40 ft/s. s(t) = so+vot-16t2
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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)
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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) y’(cos x) = - sin x
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Example
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Example
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Simple Harmonic Motion
A weighted spring is an example of simple harmonic motion Its motion is modeled by cos.
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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.
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Other trig function rules
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Example Verify the formula
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Example Find the tangent line to the function below at x=2.
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Example Find y” if y = sec x
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Assignment Pg. 165: # 1-17 odd Pg. 170: # 1-33 odd, all
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