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AP Calculus BC AP Calculus BC 7.4 Reasoning Using Slope Fields Exam Style Questions – FRQ Exam Style Questions – FRQ

AP Calculus BC 7.4 Reasoning Using Slope Fields Exam Style Questions - FRQ

Question

(a)-Topic-7.3 Sketching Slope Fields

(b)-Topic-7.4 Reasoning Using Slope Fields

(c)-Topic-7.8 Exponential Models with Differential Equations

(d)-Topic-7.6 Finding General Solutions Using Separation of Variables

3. A bottle of milk is taken out of a refrigerator and placed in a pan of hot water to be warmed. The increasing function M models the temperature of the milk at time t, where M(t) is measured in degrees Celsius (°C) and t is the number of minutes since the bottle was placed in the pan. M satisfies the differential equation \(\frac{dM}{dt}=\frac{1}{4} (40-M)\). At time t = 0, the temperature of the milk is 5°C. It can be shown that M(t) < 40 for all values of t.

(a) A slope field for the differential equation \(\frac{dM}{dt}=\frac{1}{4} (40-M)\) is shown. Sketch the solution curve through the point (0, 5).

(b) Use the line tangent to the graph of M at t = 0 to approximate M(2), the temperature of the milk at time t = 2 minutes.

(c) Write an expression for \(\frac{d^{2}M}{dt^{2}}\) in terms of M . Use\(\frac{d^{2}M}{dt^{2}}\) to determine whether the approximation from part (b) is an underestimate or an overestimate for the actual value of M(2). Give a reason for your answer.

(d) Use separation of variables to find an expression for M(t), the particular solution to the differential equation \(\frac{dM}{dt}=\frac{1}{4} (40-M)\) with initial condition M(0) = 5.

▶️Answer/Explanation

3(a) A slope field for the differential equation \(\frac{dM}{dt}=\frac{1}{4} (40-M)\) is shown. Sketch the solution curve through the point (0, 5) .

3(b) Use the line tangent to the graph of M at t = 0 to approximate M (2 ,) the temperature of the milk at time t = 2 minutes.

\(\frac{dM}{dt}|_{t=0}=\frac{1}{4}(40-5)=\frac{35}{4}\)
The tangent line equation is \(y=5+\frac{35}{4} (t-0)\)
\(M(2) \approx 5+\frac{35}{4}.2=22.5\)

The temperature of the milk at time t = 2 minutes is approximately 22.5\(^{\circ}\) Celsius.

3(c) Write an expression for\(\frac{d^{2}M}{dt^{2}}\) in terms of M. Use\(\frac{d^{2}M}{dt^{2}}\)  to determine whether the approximation from part (b) is an underestimate or an overestimate for the actual value of M (2 .) Give a reason for your answer.

\(\frac{d^{2}M}{dt^{2}}=\frac{1}{4}\frac{dM}{dt}=-\frac{1}{4}\left ( \frac{1}{4}(40-M) \right )=-\frac{1}{16}(40-M)\)
Because \(M(t) < 40,\frac{d^{2}M}{dt^{2}}< 0\) so the graph of M is concave down. Therefore, the tangent line approximation of M (2) is an overestimate.

3(d) Use separation of variables to find an expression for M t( ), the particular solution to the differential equation \(\frac{dM}{dt}=\frac{1}{4} (40-M)\)  with initial condition M (0 5).

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