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Midterm

Math 102 - Midterm 1 (2009) Solutions.pdf

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Department
Mathematics
Course
MATH 102
Professor
Yue Xian Li
Semester
Winter

Description
Midterm Examination 1 — Solutions Mathematics 102 Section 106 — October 9, 2009 1. A cart travels along a track. From t = 0 seconds to t = 4 seconds, the position of the cart is given by s(t) = 1t + t − 3t + 7 in metres. What is the average rate of change of the 4 3 acceleration of the cart over this time period? 0 5 4 2 00 3 2 The velocity is v(t) = s (t) = 4t − 8t m/s. The acceleration is a(t) = s (t) = 5t − 16t m/s . The a(4)−a(0) 3 average rate of change of the acceleration over the interval [0,4] is 4−0 s = 64 m/s . 3 2 2. Let C be the curve y = 4x + 7x − 3 and L be the tangent line to C at x = 0. At which points do L and C intersect? Let f(x) = 4x + 7x − 3, so f (x) = 12x + 14x. Note that f (0) = 0, so the slope of L is 0. We find 3 2 that f(0) = 4(0) + 7(0) − 3 = −3 so L goes through the point (0,−3). Thus the y-intercept is −3. The equation of L is y = 0x − 3, or just y = −3. Where L and C intersect they must have the same x and y values. Equating y’s we get 4x + 7x − 3 = −3. Solving this equation gives x = 0 and x = − . 7 4 Since both points lie on L (with equation y = −3), their y-values are −3. Thus L and C intersect at the points (0,−3) and (− ,−3). 4 4 2 3. Find all values of r where the graph of the function h(r) = r − r − 7 has local extrema or inflection points. Is the function h(r) even, odd or neither? First, we find h (r) = 4r − 2r. Solving h (r) = 0 gives critical points at r = 0 and r = ± √1 . To find 2 whether or not they are critical points we will use the second derivative
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