MAT2500-02 08S Quiz 6 Print Name (Last, First) _________________________________|__Show all work, including mental steps, in a clearly organized way that speaks for itself. Use proper mathematical notation, identifying expressions by their proper symbols (introducing them if necessary), and use arrows and equal signs when appropriate. Always simplify expressions. BOX final short answers. LABEL parts of each problem. Keep answers EXACT (but give decimal approximations for interpretation when appropriate). Indicate where technology is used and what type (Maple, GC).
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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUYsNiVRInhGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRInlGJ0YvRjJGQUZBLUY+Ni1RIn5GJ0ZBRkMvRkdGRUZIRkpGTEZORlBGUi9GVkZULUY+Ni1RIj1GJ0ZBRkNGaG5GSEZKRkxGTkZQL0ZTUSwwLjI3Nzc3NzhlbUYnL0ZWRl5vLUYjNiotRiw2I1EhRictRiM2Ji1JI21uR0YkNiRRIjRGJ0ZBLUY+Ni1RMSZJbnZpc2libGVUaW1lcztGJ0ZBRkNGaG5GSEZKRkxGTkZQRlJGaW4tSSVtc3VwR0YkNiVGOi1GaG82JFEiMkYnRkEvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRkEtRj42LVEoJm1pbnVzO0YnRkFGQ0ZobkZIRkpGTEZORlAvRlNRLDAuMjIyMjIyMmVtRicvRlZGW3EtRiM2KEZhcEZbcEY6RltwRlhGQS1GPjYtUSIrRidGQUZDRmhuRkhGSkZMRk5GUEZqcEZccS1GIzYnLUkmbWZyYWNHRiQ2KC1GaG82JFEiNUYnRkFGYXAvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRl9yLyUpYmV2ZWxsZWRHRkVGW3AtRiM2JC1GX3A2JUZYRmFwRmRwRkFGYm9GQUZib0ZBRmJvRkE=a) Put a circled dot at the point (1,2) and draw in your guess of the tangent line and normal line to the level curve of the function through that point, using a straight edge.b) Evaluate the gradient vector field and its value at (1,2), identifying it with its proper symbol, and draw its direction unit vector into your diagram.c) Write an equation for the normal line to the level curve through (1,2). What is its y intercept? Does it seem consistent with your hand drawn line?d) Write an equation for the tangent line to this level curve. What is its y intercept? Does it seem consistent with your hand drawn line?
e) What is the derivative of f at (1,2) in the direction of the point (0,1)? Identify it by its proper symbol and include the associated direction unit vector 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 in the diagram. Is the function increasing or decreasing in this direction?f) Write an equation for the level curve at (1,2).
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QyU+SSJmRzYiZio2JEkieEdGJUkieUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCgqJDkkIiIjIiIlKiZGLyIiIjklRjMhIiMqJEY0RjAjIiImRjBGJUYlRiVGMy1GJEYnb)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e)The difference vector and its direction 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directional derivative is the dot product of the gradient with the unit vector direction:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIi5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSdncmFkZjBGJ0YvRjIvJStleGVjdXRhYmxlR0Y9Rjk=b) the direction vector of the gradient itself gives the direction of maximum increase of the function, while its magnitude gives the rate of change in that direction: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c) d)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 tangent line is perpendicular to the gradient (slope 2), so has slope 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technology we might as well add in the level curve through the point 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