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2. (18 pts) A part of the curve with equation is sketched below. cos (xy) + x + y = 1 13 0.6 0.7 0.8 0.9 1.0 1.1 1. (a) Use implicit differentiation to find the derivative (b) Consider the point (1, 1). Show (algebraically) that this point lies on the curve. (c) Find the equation of the line tangent to the curve at (1, 1). Draw this line in the figure above.

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30 Nov 2018
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5. (18 pts) A curve is given by the equation tan-(y) + x + y In(x - 1) = 2. The questions (a) - (e) below are about this curve. (a) (10 pts) Find y, using implicit differentiation. Show your work. . (b) (2 pts) Verify that the point (2,0) lies on this curve. 1 2 3 4 5 6 ) (2 pts) A part of the curve that contains a point (2,0) is shown in the figure. In the figure above, draw the line tangent to the curve at the point (2,0). (d) (2 pts) Find the slope of the line tangent to the curve at the point (2,0). (e) (2pts) Write an equation of the line tangent to the curve at the point (2,0).

1. (30 pts) (I) Calculate the derivatives of the following functions. NO NEED TO SIMLIFY ! (a) (8 pts) y = x tan-1 (x5) (b) (8 pts) y = xsin x (II) (14 pts) The curve with equation is a "rotated ellipse". x2 + xy + y2 = 4 (a) Use implicit differentiation to find the derivative (b) Consider the two points (3, -1) and (2, -2). Show that one of these points lies on the ellipse defined above, and one does not. (c) For the point in part (b) which lies on the ellipse, find the equation of the line tangent to the ellipse at this point.

3. (15 pts) (I) The graph of the functions (t) =- - t +1 is given below. s=s(t) (a) In the figure above draw a secant line joining the points (o, s (0)) and (2, s (2)). (b) Find the slope, msec, of this secant line. (c) Draw a tangent line to the curve s = s(t) at the point where t = 2. (d) Is the slope of this tangent line, mtan, positive or negative ? 12 - is the position of an object moving along a line (II) Suppose s (t) = — at time to. (a) How is the average velocity, Vav, between the times t = 0 and t = 2 related to the slope msec from the part (I b)? (b) Is the instantaneous velocity, Vinst, at t = 2 positive or negative? Explain.

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