Parallelogram is rotated to create image . On a coordinate plane, parallelograms are shown. The first parallelogram has points , , ,.The second parallelogram has points , ,,. Which rule describes the transformation?
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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.
In this exercise, we find a formula for the angle formed by two lines in a coordinate plane.
(a) If is a line in the plane and is the angle formed by the line and the -axis as shown in the figure, show that the slope of the line is given by
(b) Let and be two nonparallel lines in the plane with slopes and , respectively. Let be the acute angle formed by the two lines (see the following figure). Show that
(c) Find the acute angle formed by the two lines
and
(d) Show that if two lines are perpendicular, then the slope of one is the negative reciprocal of the slope of the other. [Hint: First find an expression for .]
