Question

勉強しろ自分

赤く丸をしたbの問題で解答の方に二階微分した後の式がなぜ(-1/4)(-1/4)(H-27)になるのか分かりません。教えてください🙇‍♀️

QA
At time t = 0, a boiled potato is taken from a pot on a stove and left to cool in a kitchen. The internal
temperature of the potato is 91 degrees Celsius (°C) at time t = 0, and the internal temperature of the potato
is greater than 27°C for all times t > 0. The internal temperature of the potato at time t minutes can be
modeled by the function H that satisfies the differential equation
dH
(H-
(H-27), where H(t) is
dt
measured in degrees Celsius and H(0) = 91.
(a) Write an equation for the line tangent to the graph of Hat t = 0. Use this equation to approximate the
internal temperature of the potato at time t = 3.
(b) Use
2017 APⓇ CALCULUS AB FREE-RESPONSE QUESTIONS
(a) dH
d²H
dt²
to determine whether your answer in part (a) is an underestimate or an overestimate of the
internal temperature of the potato at time t = 3.
(c) For t < 10, an alternate model for the internal temperature of the potato at time 7 minutes is the function
-= − (G - 27)²/3, where G(t) is measured in degrees Celsius
dG
G that satisfies the differential equation
dt
and G(0) = 91. Find an expression for G(t). Based on this model, what is the internal temperature of the
potato at time t = 3 ?
564
at (21-27)
-
==
2-16
To = - = (H(3)-27)
4
-64 = HB)-27
-37 = H (3)
(b) _d²fi
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(c)
H
(a) H'(0) = -(91-27) = -16
H(0) = 91
An equation for the tangent line is y = 91 - 16t.
The internal temperature of the potato at time 1 = 3 minutes is
approximately 91-16-3 = 43 degrees Celsius.
(b)²-1(1)(-1)(-27)=(-27)
dH
4 dt
dG
(G-27)²/3
AP® CALCULUS AB/CALCULUS BC
2017 SCORING GUIDELINES
= −dt
H>27 for t> 0 ⇒
d² H
=(H-27) > 0 for t > 0
dt²
Therefore, the graph of H is concave up for t > 0. Thus, the
answer in part (a) is an underestimate.
dG
√(G-2973²/3 = S(-1) dr
Question 4
3(G-27)¹/³ = -t + C
3 (91-27)¹/3 = 0+C ⇒ C = 12
3(G-27)¹/3 = 12-1
G(t) = 27 +
for 0 ≤ 1 <10
(H
The internal temperature of the potato at time t = 3 minutes is
= 54 degrees Celsius.
27 +
1: slope
3: 1: tangent line
1: approximation
1: underestimate with reason
5:
1: separation of variables
1: antiderivatives
1: constant of integration and
uses initial condition
1: equation involving G and t
1: G(t) and G(3)
Note: max 2/5 [1-1-0-0-0] if no const
of integration
Note: 0/5 if no separation of variable:
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Visit the College Board on the Web: www.collegeboard.org.