As a pot of tea cools, the temperature of the tea is modeled by a differentiable function $H$ for $0 \leq t \leq 10$, where time $t$ is measured in minutes and temperature $H ( t )$ is measured in degrees Celsius. Values of $H ( t )$ at selected values of time $t$ are shown in the table above. (a) Use the data in the table to approximate the rate at which the temperature of the tea is changing at time $t = 3.5$. Show the computations that lead to your answer. (b) Using correct units, explain the meaning of $\frac { 1 } { 10 } \int _ { 0 } ^ { 10 } H ( t ) d t$ in the context of this problem. Use a trapezoidal sum with the four subintervals indicated by the table to estimate $\frac { 1 } { 10 } \int _ { 0 } ^ { 10 } H ( t ) d t$. (c) Evaluate $\int _ { 0 } ^ { 10 } H ^ { \prime } ( t ) d t$. Using correct units, explain the meaning of the expression in the context of this problem. (d) At time $t = 0$, biscuits with temperature $100 ^ { \circ } \mathrm { C }$ were removed from an oven. The temperature of the biscuits at time $t$ is modeled by a differentiable function $B$ for which it is known that $B ^ { \prime } ( t ) = - 13.84 e ^ { - 0.173 t }$. Using the given models, at time $t = 10$, how much cooler are the biscuits than the tea?
: \text { integrand }
As a pot of tea cools, the temperature of the tea is modeled by a differentiable function $H$ for $0 \leq t \leq 10$, where time $t$ is measured in minutes and temperature $H ( t )$ is measured in degrees Celsius. Values of $H ( t )$ at selected values of time $t$ are shown in the table above.
(a) Use the data in the table to approximate the rate at which the temperature of the tea is changing at time $t = 3.5$. Show the computations that lead to your answer.
(b) Using correct units, explain the meaning of $\frac { 1 } { 10 } \int _ { 0 } ^ { 10 } H ( t ) d t$ in the context of this problem. Use a trapezoidal sum with the four subintervals indicated by the table to estimate $\frac { 1 } { 10 } \int _ { 0 } ^ { 10 } H ( t ) d t$.
(c) Evaluate $\int _ { 0 } ^ { 10 } H ^ { \prime } ( t ) d t$. Using correct units, explain the meaning of the expression in the context of this problem.
(d) At time $t = 0$, biscuits with temperature $100 ^ { \circ } \mathrm { C }$ were removed from an oven. The temperature of the biscuits at time $t$ is modeled by a differentiable function $B$ for which it is known that $B ^ { \prime } ( t ) = - 13.84 e ^ { - 0.173 t }$. Using the given models, at time $t = 10$, how much cooler are the biscuits than the tea?