The values of certain types of collectibles can often fluctuate greatly over tim
ID: 3123280 • Letter: T
Question
The values of certain types of collectibles can often fluctuate greatly over time. Suppose that the value of a particular limited edition ceramic Donald Trump garden gnome is found to be able to be modeled by the function V(t) = -0.01t4 + 0.042t3 - 6.43t2 + 40.78t + 59 for 0 t 20 where V(t) is in dollars, t is the number of years after the gnome was released, and t= 0 corresponds to the year 1996.
a) what was the value of the gnome in the year 2001?
2016
what was the instantaneous rate of change of the value of the gnome in 1999
instantaneous rate of change of the value of the gnome in the year 2010
E0 Use answers from parts a-d to ESTIMATE the value of the gnome in 2017?
Explanation / Answer
A)
V(t) = -0.01t4 + 0.042t3 - 6.43t2 + 40.78t + 59 for 0 t 20 where V(t) is in dollars, t is the number of years after the gnome was released, and t= 0 corresponds to the year 1996.
for the year 2001, t = 5.
using t = 5 in equation we get
V(5) = -0.01*5^4 + 0.042*5^3 - 6.43*5^2 + 40.78*5 + 59
= -6.25 + 5.25 - 160.75 + 203.9 + 59
= 101.15
Value of gnome, V(t) = 101.15 for year 2001.
B)
Instantaneous Rate of Change Definition
Instantaneous rate of change is the interpretation of the derivative of a function at a point. The derivative is the limit of the average rate of change between a fixed point and the other point of the curve which is closer to that point(fixed point). If for a function, limit exists, then it is defined as the instantaneous rate of change at that fixed point.
Instantaneous rate of change is the rate of change at that moment. For example, the value of a function at a point x=ax=a is f(a)f(a). If there is a small increment in xx, which is xx, then the value of the function at a+xa+x is f(a+x)f(a+x). Hence, the change in the value of the function is f(a+x)f(a)f(a+x)f(a). By the term difference of quotient, we mean the change in the value of the function over the change in the value of xx which is xx.
Instantaneous Rate of Change Formula
The instantaneous rate of change of the function y=f(x)y=f(x), with respect to xx, when x=ax=a is given by the formula,
f(a)=limx0f(a)=limx0yxyx = limx0limx0t(a+h)(t(a)ht(a+h)(t(a)h provided the limit exists.
It should be noted that the instantaneous rate of change of y=f(x)y=f(x) with respect to xx, when x=ax=a is the slope of the tangent at x=ax=a.
V(t) = -0.01t4 + 0.042t3 -6.43t2 + 40.78t +59 -------- eq1
for year 1999, t=3
V(3) = 123.794
V'(t) = Limx->0( V(3+h)-V(3) ) / h -------- eq2
V(3+h) = -0.01*(3+h)^4 + 0.042*(3+h)^3 - 6.43^(3+h)^2 + 40.78*t +59
putting V(3+h) in eq2, we get
Instataneous rate of change=(-0.01[108h + 54h2 + 12h3 + h4] 0.042[27h + 9h2 + h3] - 6.43[6h + h2] + 40.78[h])/h
taking h common from the numerator and putting h = 0 in eq
we get -1.08 + 1.134 - 38.58 + 40.78 = 2.254
Instantaneous rate of change of the value of gnome in year 1999 is 2.254
C) Similarly solving instantaneous rate of change in value of gnome in year 2010 using t = 14
PS: you don't have to solve the eqn again. just take out the first derivatives of h (all values with h and not h^2 etc.)
just compute the h part and you will get the answer.
Instantaneous rate of change of the value of gnome in year 2010 is -224.324
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