Unformatted text preview: We know that, given a differentiable function , that the line tangent to the curve at , and a point on the graph has slope and equation If is "sufficiently close" to , then we can use points on the tangent line as approximations to . In this sense, the tangent line is called "the linearization of 6. linearization of ln at at ." ln linearization is MTH 173, section 3.11 notes, page 1 linear approximation of and Graph at and use it to approximate tangent line at Since approximate so Similarly, so , from Derive , from Derive , so , we see that by determining the , so we coordinate on the tangent when MTH 173, section 3.11 notes, page 2 12. verify tan to within tan tan at . Determine interval in which approximation is accurate sec sec with a slope of : linearization is line thru MTH 173, section 3.11 notes, page 3 since the linearization of tan at is near , we have tan tan We've seen the notation in . If , then We now define: and if representing a change in , and representing a change the differential , the differential MTH 173, section 3.11 notes, page 4 16. differential of cos sin 22. differential of evaluate for red: curve; blue: tangent 28. given compute and . Sketch MTH 173, section 3.11 notes, page 5 40. linearizations at . ln ln all linearizations have slope of and go thru , so they are MTH 173, section 3.11 notes, page 6 red: blue: black: 42 a. here radius of circular disk is 24 cm with max error in measurement of 0.2 cm. use differentials to estimate the maximum error in caluclated area of disk. determine
MTH 173, section 3.11 notes, page 7 cm calculated area b. cm what is the relative error? percent error?
error calculated value relative error percent error relative error ! % 44.Use differentials to estimate amount of paint required to apply coat 0.05 cm thick to hemispherical dome of diameter 50 m. volume of sphere volume of hemisphere " since cm ! cm liters of paint liter, this paint job requires about MTH 173, section 3.11 notes, page 8 ...
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