Euler's Method for Approximating Functions
Learn how to use Euler's method for approximating functions when direct solutions to differential equations are not possible. Discover the numerical techniques and steps involved in estimating values of functions at specific points. Explore the concept of tangent lines and their role in the approximation process.
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Presentation Transcript
Section 2.7 Eulers method: Using tangent lines to approximate a function. We often cannot solve differential equations directly and thus need to use numerical methods to approximate solutions. For more info (and to earn a math minor), take MATH:3800 Introduction to Numerical Methods. Math minor info (15 math units): https://math.uiowa.edu/undergraduate- program/majors-minor/minor-requirements Or double major: eg Math Program C: Engineering https://math.uiowa.edu/sites/math.uiowa.edu/files/p rint_pdf/Engineering%20Program%20C%20Template% 202016%20.pdf http://slopefield.nathangrigg.net
?? + 1 = ?? + ? = ??+ ? ??,?? ? Part of tangent line at (??,??) Thus slope is ?(??,??) (??+ 1,??+ 1) Its length is determined by ? ? (??,??) ? = 0.1 (t2, y2) (t1, y1) (t3, y3) (t0, y0)
Estimate y(0.3) using h = Step size = t = 0.1 Thus if (?0,?0) = (0,0) the initial value, then (?? + 1,??+ 1) ?1 = 0.1 and ?1 = ?0 + ? = ?0+ ? ?0,?0 0.1 ? Slope at (?0,?0) ? (??,??) ? = 0.1 ?2 = 0.2 and ?2 = ?1 + ? = ?1+ ? ?1,?1 ?3 = 0.3 and ?3 = ?2 + ? = ?2+ ? ?2,?2 (0.1) (0.1)
h = Step size = t = 0.1
For the initial value problem, y = t2y, y(5) = 3, use Euler s method to estimate y(5.2) using t = 0.1
h = Step size = t = 0.1 Sequence of functions One function consisting of pasting together tangent lines
2.7: Glue together tangent lines to find a single function approximating y(t) h = Step size = t = 0.1 2.8: Find a sequence of functions that converge to y(t)
