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Investigate the given two parameter family of functions. Assume that a and b are positive. f(x) = a / x + bx for x > 0 Graph f(x) using b = 1 and three different values for a. Graph f(x) using a = 1 and three different values for b. In the graph in part (a), how do the critical points of f appear to move as a increases? The critical point moves down and to the left as a increases. The critical points move down and to the right as a increases. The critical point does not move as a increases. The critical points move up and to the left as a increases. The critical point moves up and to the right as a increases. Find the formula for the x-coordinate of the critical point(s) of f in terms of a and b.(Enter your answers as a comma-separated list.) x = If a and b are positive constants, find all critical points of h(t) = at2 / t - b. t = (smaller value) This critical point is a t = (larger value) This critical point is a . If a and b are positive constants, find the critical point of g(w) = a / w3 - b / w2 where w > 0. W = This critical point is a . A family of functions is given by r(x) = 1 / a - (x - b)2. Assume a > 0, b > 0. Find the critical point of r(x). x = Classify the critical point as a minimum, maximum, or neither. maximum neither minimum Find the vertical asymptotes of r(x). x = (smaller value) x = (larger value) A family of functions is given by r(x) = 1 / a - (x - b)2. Assume a > 0, b > 0. Find the critical point of r(x). x = Classify the critical point as a minimum, maximum, or neither. maximum neither minimum Find the vertical asymptotes of r(x). x = (smaller value) x = (larger value) The temperature , T, in degree C, of a yam put into a 275 degree C oven is given as a function of time, t, in minutes, by T = a(1 - e -kt) + b. a = b = If the temperature of the yam is initially increasing at 4 degree C per minute, find k. k =
Explanation / Answer
1)
a)a
because f is always positive and x>0
b) d
c) b in part (a) and c in part (b)
d) square root (-ab)
2)
h(t) has local min at t=0 and local max at t=2b
3)
at w= 3a/2b local max exists and is the critical point
5)
at x= b is the critical point and local max occurs
and the asymptotes are : largest value is 1/a and smallest is -infinity
6)
a)the given function becomes y axis if y=0 so the x intercept is the at origin
b)a
c) a can be any value
critical point is origin and it is neither local min or max
7) b)k=ln((4-b /a) +1)
