m2121t1rev-sol - MATH/2121 CALCULUS FOR THE LIFE SCIENCES I...

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MATH/2121, CALCULUS FOR THE LIFE SCIENCES I TEST 1 REVIEW PROBLEMS SOLUTIONS (1) What is the slope of the line through ( - 2 , 1) and (6 , 3)? m = 3 - 1 6 - ( - 2) = 2 8 = 1 4 . (2) Give equations for the following lines. (a) The line with slope -2 and y -intercept (0 , 4): y = - 2 x + 4 . (b) The line through (1 , - 1) with slope 2: y = 2( x - 1) + ( - 1) = y = 2 x - 3 (c) The line through (2 , 4) and (5 , 3): The line has slope m = (3 - 4) / (5 - 2) = - 1 / 3 and equation y = - 1 3 ( x - 2) + 4 = y = - 1 3 x + 14 3 . (d) The horizontal line through ( - 1 , 3): y = 3 . (e) The vertical line through ( - 1 , 3): x = - 1 . 1
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(3) Find the vertex and sketch the graph for the following quadratic polynomials. (a) y = 3 x 2 - 2 x + 1 Vertex has x -coordinate - b/ 2 a = - ( - 2) / (2 · 3) = 1 / 3 and y -coordinate 3(1 / 3) 2 - 2(1 / 3) + 1 = 2 / 3. So we have vertex 1 3 , 2 3 . (b) y = - 2 x 2 - 4 x + 6 Vertex is ( - 1 , 8) . (4) What are the y -intercepts for the parabolas in #3? What are the x -intercepts, if any? (a) y -intercept is (0 , 1) . There are no x -intercepts. (See graph and note that b 2 - 4 ac = - 8 < 0.) 2
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(b) y -intercept is (0 , 6) and we have x -intercepts ( - 3 , 0) , (1 , 0) . (See graph and note that - 2 x 2 - 4 x + 6 = - 2( x 2 + 2 x - 3) = - 2( x + 3)( x - 1) has roots x = - 3 , 1. Or resort to the quadratic formula .) (5) Sketch the graphs for each pair of equations, using a common set of axes. (a) y = x and y = x + 1. (b) y = x and y = 4 - x . (c) y = x and y = 2 - x . 3
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(d) y = x and y = x - 1 - 2 (In each plot the graph for y = x is shown on black and the trans- lated/reflected function shown in red.) (6) Determine the domain of the function. Express your answers using interval notation. (a) f ( x ) = x 4 + 7 x 2 + 6 x + 5 .
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