Question:
The temperature of a liquid is changing with respect to time. The formula that models the change in temperature (in degrees Celsius) over time (in minutes) is given by:
T(t)=2t3−5t2+8t−3Find the linear approximation of the temperature at t=2 minutes. Use this approximation to estimate the temperature at t=2.1 minutes. Round your answer to two decimal places.
Answer:
To find the linear approximation of the temperature at t=2 minutes, we will use the concept of the tangent line to approximate the temperature near that point. The equation of the tangent line is given by:
L(t)=f(a)+f′(a)⋅(t−a)where f(a) represents the value of the function at t=a, and f′(a) represents the derivative of the function evaluated at t=a. In our case, a=2.
First, let's find the value of the temperature at t=2 minutes:
T(2)=2(2)3−5(2)2+8(2)−3=16−20+16−3=9So, f(2)=9.
Now, let's find the derivative of the temperature function:
T′(t)=dtd(2t3−5t2+8t−3)Applying the power rule and the sum rule of derivatives, we get:
T′(t)=6t2−10t+8Next, we need to evaluate the derivative at t=2:
T′(2)=6(2)2−10(2)+8=24−20+8=12So, f′(2)=12.
Now, let's substitute these values into the equation of the tangent line to find the linear approximation at t=2 minutes:
L(t)=f(2)+f′(2)⋅(t−2)L(t)=9+12⋅(t−2)Simplifying the expression:
L(t)=−3+12tTherefore, the linear approximation of the temperature at t=2 minutes is L(t)=−3+12t.
To estimate the temperature at t=2.1 minutes using this linear approximation, we substitute t=2.1 into the equation of the tangent line:
L(2.1)=−3+12(2.1)L(2.1)=−3+25.2L(2.1)=22.2Therefore, the estimated temperature at t=2.1 minutes is approximately 22.2 degrees Celsius.