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A math exam focused on calculus concepts such as finding gradients, critical points, tangent lines, and approximating functions using taylor polynomials. The exam covers functions of two variables and includes problems involving finding derivatives, classifying critical points, and sketching vectors. It also includes problems on vector calculus, specifically finding the divergence and curl of a vector field.
Typology: Exams
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1b) Find all the critical points of f.
1c) Classify each of the critical points in problem (1b) as local max/mins/saddle points according to the second derivative test. If the test fails, explain why. ORGANIZE your answers neatly.
2A) Find ∇f.
2B) There’s a dot at (4,1). In the form y = mx + b, find the equation of the line tangent to the level curve there. Show all your work. Then make an excellent sketch, with the right slope, of this line on your graph.
2C) Find a unit vector in the direction of the gradient at the point (4,1). Accurately sketch this vector on the graph, with its “tail” at (4,1).
2D) There appears to be a local maximum at some point (p, 0) on the x-axis to the right of (3, 0). Use ∇f to find the value of p exactly.
2E) What is the equation of the plane tangent to the graph of f at this local maximum (ie, at (p, 0 , f(p, 0))? (this is an easy question: think about what the plane must look like at a maximum)
4A. Find the second order Taylor polynomial for f(x, y) = y
x for the point a = (25, 3).
4B) Use that polynomial from 4A to approximate 3. 02
26 (that is, f(26, 3 .02)). Compare the approximation to the actual (calculator) value by finding the differences.
5A: the divergence is:
5B the curl is:
