Drill: Find dy / dx

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# Drill: Find dy / dx - PowerPoint PPT Presentation

Drill: Find dy / dx. y = x 3 sin 2x y = e 2x ln (3x + 1) y = tan -1 2x Product rule: x 3 (2cos 2x) + 3x 2 sin (2x) 2x 3 cos 2x + 3x 2 sin (2x). Product Rule e 2x (3/(3x +1)) + 2e 2x ln (3x + 1) 3e 2x /(3x +1) + 2e 2x ln (3x + 1). Antidifferentiation by Parts. Lesson 6.3.

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Presentation Transcript
Drill: Find dy/dx
• y = x3 sin 2x
• y = e2xln (3x + 1)
• y = tan-1 2x
• Product rule:
• x3 (2cos 2x) + 3x2 sin (2x)
• 2x3cos 2x + 3x2 sin (2x)
• Product Rule
• e2x (3/(3x +1)) + 2e2xln (3x + 1)
• 3e2x/(3x +1) + 2e2xln (3x + 1)

### Antidifferentiation by Parts

Lesson 6.3

Objectives
• Students will be able to:
• use integration by parts to evaluate indefinite and definite integrals.
• use rapid repeated integration or tabular method to evaluate indefinite integrals.
Integration by Parts Formula

A way to integrate a product is to write it in the form

If u and v are differentiable function of x, then

Example 3 Solving an Initial Value Problem
• Solve the differential equation dy/dx = xlnx subject to the initial condition y = -1 when x = 1

It is typically better to let u = lnx

Drill

Solve the differential equation: dy/dx= x2e4x (This means you will need to find the anti-derivative of dy/dx = x2e4x)

Rapid Repeated Integration by PartsAKA: The Tabular Method
• Choose parts for u and dv.
• Differentiate the u’s until you have 0.
• Integrate the dv’s the same number of times.
• Multiply down diagonals.
• Alternate signs along the diagonals.
Homework
• Page 346/7: Day #1: 1-15 odd
• Page 347: 17-24