The Detective s Hat Function
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1 The Detective s Hat Function (,) (,) (,) (,) (, ) (4, ) The graph of the function f shown above is a piecewise continuous function defined on [, 4]. The graph of f consists of five line segments. Let g be the function given by g ( ) = ftdt ( ).. Find each of the following. (a) g( ) (b) g( ) (c) g() (d) g() (e) g() (f) g(4). Eplain the procedure you followed to answer question.. Find each of the following. (a) g ( ) (b) g () (c) g () (d) g (4) 4. Eplain the procedure you followed to answer question. 5. Eplain why g must be a continuous function on [, 4]. 6. Write the equation for g ( ) on the interval [, ]. 7. Write the equation for the line tangent to g at =. Justify your answer. 8. Does g () eist? Eplain your 9. Will a point of inflection for g eist when =? Eplain your. For what values of in the open interval (, 4) is g increasing? Eplain your. For what values of in the open interval (, 4) is g decreasing?
2 . For what values of in the open interval (, 4) is g concave up? Eplain your. For what values of in the open interval (, 4) is g concave down? 4. Find the maimum and the minimum values of g on the closed interval [, 4]. Justify your answers. 5. On the aes provided, sketch the graph of function g on the closed interval [, 4]. y For questions 5 7, let h be the function given by ( ) ( ) h = f tdt. 6. Find each of the following. (a) h( ) (b) h( ) (c) h() (d) h() (e) h() (f) h(4) 7. Find the following and eplain your (a) h ( ) (b) h () (c) h () (d) h (4) 8. g ( ) h ( ) = k, where k is a constant. Find the value of k and eplain your 9. If w ( ) = f( tdt ), find w().. w() can also be defined as value of r? w ( ) = r+ f( tdt ) where r is a constant. What is the
3 The Detective s Hat Function Answers (,) (,) (,) (,) (, ) (4, ) The graph of the function f shown above is a piecewise continuous function defined on [, 4]. The graph of f consists of five line segments. Let g be the function given by g ( ) = ftdt ( ).. Find each of the following. (a) g( ) = (b) g( ) = (c) g() = (d) g() = 5 (e) g() = 6.5 (f) g(4) = 6. Eplain the procedure you followed to answer question. The definite integrals can be calculated using geometry. f () is the sum of two triangles. ( )( ) + ( )() =. The other values are calculated in a similar fashion.. Find each of the following. (a) g ( ) = (b) g () = (c) g () = (d) g (4) = 4. Eplain the procedure you followed to answer question. Since the derivative of g is equal to, the values of the derivative are the same as the values of f
4 5. Eplain why g must be a continuous function on [, 4]. Since g ( ) eists on the interval [, 4] g must be continuous on the interval. 6. Write the equation for g ( ) on the interval [, ]. g ( ) = + 7. Write the equation for the line tangent to g at =. Justify your answer. 9 g () =, 4 5 g () =, 5 9 y = ( ) Does g () eist? Eplain your No, g has a cusp at ; therefore, g () will not eist. (Or show that the right and left limits of g are not the same.) 9. Will a point of inflection for g eist when =? Eplain your Since g is increasing on both sides of, there will not be a point of inflection at =.. For what values of in the open interval (, 4) is g increasing? Eplain your g ( ) g is increasing for < < where g >. + g ( ) dec inc dec 4. For what values of in the open interval (, 4) is g decreasing? g is decreasing for < < and again when < < 4 where g <.. For what values of in the open interval (, 4) is g concave up? Eplain your g will be concave up when g is increasing and where g ( ) eists and is not equal to. g is concave up for (, ), (, ), and (, ). (Note: A function is concave up if the second derivative is positive and concave down when the second derivative is negative. No concavity eists when the second derivative is. This is why a straight line is neither concave up or down. 4
5 . For what values of in the open interval (, 4) is g concave down? g will be concave down when g is deceasing and where g ( ) eists and is not equal to. g is concave down for (, ), and (, 4). 4. Find the maimum and the minimum values of g on the closed interval [, 4]. Justify your answers. g ( ) + g ( ) dec inc dec 4 The maimum value must occur at = or at =. g( ) = therefore, the maimum value of g is 6.5. and g() = 6.5; The minimum value must occur at = or at = 4. g( ) = and g(4) = 6; therefore, the minimum value of g is. 5. On the aes provided, sketch the graph of function g on the closed interval [, 4]. (This function is concave up on the intervals (, ), (, ), and (, ); however, on the interval (, ) the curvature is so slight that the function appears to be linear) The slope of the tangent line at = is. 5
6 For questions 5 7, let h be the function given by ( ) ( ) h = f tdt. (a) h( ) = (b) h( ) = (c) h() = (d) h() = (e) h() = 7 (f) h(4) = 7. Find the following and eplain your (a) h ( ) = (b) h () = (c) h () = (d) h (4) = 8. g ( ) h ( ) = k, were k is a constant. Find the value of k and eplain your Since g and h have the same derivative, they can differ at most by a constant. That constant is ½. Or k = k 9. If = f () + = = = w ( ) = f( tdt ), find w().. w() can also be defined as value of r? 6 = = 6 w ( ) = r+ f( tdt ) where r is a constant. What is the 6
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