SPRING 2014 MATH 131 503 QUIZ 1 SOLUTION Problem 1 Find the domain and sketch the graph of the function 2 5 points x 9 if x 3 f x 2x if x 3 3 6 if x 3 Solution Clearly the given f x is piece wise continuous function the whole domain which is R is divided into three pieces we will graph these three pieces by considering the respective definition of the function For the domain 3 let us use f x x 9 which is a equation of line with slope 1 slope is the x coefficient in the given equation of f x and when is x is very close to 3 f x will be close to 6 remember we can t take x exactly equal to 3 why then we consider domain 3 3 in this domain f x 2x which is the equation of the line with slope 2 and passes through origin why finally for the domain x 3 f x 6 is a constant function The graph of the given function is as follows f x 6 5 4 3 2 1 5 4 3 2 1 1 2 3 4 5 6 1 2 3 4 5 x Quiz 1 Solution f 1 h f 1 2 5 points h Solution The given function is f x 4 3x x2 first let us find f 1 h and f 1 Problem 2 Let f x 4 3x x2 then find the value of f x 4 3x x2 f 1 h 4 3 1 h 1 h 2 f 1 h 4 3 3h 1 2h h2 f 1 h 8 5h h2 Then let us find the value of f 1 f x 4 3x x2 f 1 4 3 1 1 2 f 1 4 3 1 f 1 8 Finally using the values of f 1 h and f 1 we can find the value of f 1 h f 1 h f 1 h f 1 8 5h h2 8 h h 5h h2 h h 5 h h f 1 h f 1 5 h h Problem 3 The average surface temperature of the earth has been modeled by a linear function T 0 02t 8 5 where T is temperature in 0 C and t represents years since 1900 2 5 points a Find the slope and T intercept what do these quantities represent Slope 0 02 slope of a line is the x coefficient in the equation for line Slope represents the rate of change of surface temperature T intercept 8 5 T intercept can be obtained by taking t 0 in the given equation T intercept represents the average surface temperature in the year 1900 b Use the equation to predict the average global surface temperature in 2014 Recall t represents years since 1900 t 2014 1900 114 using t 114 in the given model we can predict the average global surface temperature in 2014 T 114 0 02 114 8 5 10 780 C 2 Spring 2014 c MSM Quiz 1 Solution Problem 4 Find an expression for the function whose graph is the given curve 2 5 points f x 1 0 x 1 Solution Clearly the given function is a piecewise function with domain 0 the first piece is defined for x 0 3 the two end points of the line are x1 y1 0 3 and x2 y2 3 0 recall that to find equation of a line we need two points on the line we then calculate the slope of the line y2 y1 0 3 1 Slope m x2 x1 3 0 Then slope point form of line is y y1 m x x1 y 0 1 x 3 y x 3 Similarly the equation of the line with the endpoints x1 y1 3 0 and x2 y2 5 4 is given by Slope m y2 y1 4 0 2 x2 x1 5 3 Then slope point form of line is y y1 m x x1 y 0 2 x 3 y 2x 6 Finally for x 5 f x 4 is a constant function Combining all the above x 3 if x 0 3 f x 2x 6 if x 3 5 4 if x 5 3 Spring 2014 c MSM
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