2. (10 pts.) (a) Differentiate: d d x [ln(x)] (b) Differentiate...

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2. (10 pts.) (a) Differentiate: d d x [ln(x)] (b) Differentiate: d d x [ln(x 2 − 4x + 2)] (c) Explain how you would differentiate d d x [ln(u)] for any expression u. 3. (10 pts.) For the relationship y 2 = x 3 − 2x + 2: (a) Find d y d x using implicit differentiation. (b) Find the slope of the tangent line to this relationship at the point (1, 1). 4. (30 pts.) Use the first and second derivative tests to sketch a graph of the function f (x) = x 3 − 3x 2 + 2x + 9. (a) Conduct the first derivative test and make a sign chart. (b) On what interval(s) is the function increasing? (c) On what interval(s) is the function decreasing? (d) Where are the local maximums and minimums of the function? (e) Conduct the second derivative test and make a sign chart. (f) On what interval(s) is the function concave up? (g) On what interval(s) is the function concave down? (h) Where are the inflection points of the function? (i) Use all of the above information to sketch a graph of f (x). −→ (The graph does not have to be to scale with the gridlines if you mark and label your points.) −10 −8 −6 −4 −2 2 4 6 8 10 −10 −8 −6 −4 −2 2 4 6 8 10 5. (10 pts.) For the function f (x) = e 2x : (a) Find d d x [ f (x)] and d 2 d x2 [ f (x)]. (b) Make a hypothesis about the value of d 10 d x10 f (x). What pattern do you see? 6. (10 pts.) Write a function which could have a second derivative of f 00(x) = 18x + 2. 7. (10 pts.) The graph below shows a function with a vertical asymptote at x = 0. For this graph: (a) Visually estimate and mark all the critical points and inflection points in the graph. (b) On what interval(s) is the function increasing? (c) On what interval(s) is the function decreasing? −10 −8 −6 −4 −2 2 4 6 8 10 −10 −8 −6 −4 −2 2 4 6 8 10 8. (10 pts.) A rectangular pasture has a river along one of its long sides. Every year 1 m of the river bank erodes away, leaving the pasture smaller than it was. If the pasture is currently 100 m by 30 m, at what rate is the area of the pasture decreasing? (The area of a rectangle is A = lw.)

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