Newton's law of cooling states that $\frac{dT}{dt}= -k(T-T_s)$ if $T(0)=100$, $T(3)=75$ and $T_s=25$ then how would you use linear approximation to estimate the value of k?
Can you use the Mean Value Theorem which tells you that at a point c $\in (0,3)$ the slope of the tangent is $\frac{-25}{3}$ ?
Here's one way to answer the question - it depends on a little bit of not quite common knowledge and may not count as linear approximation.
You know the "$1/3$ life" of the cooling process is $3$ seconds. That's how long it takes to get $1/3$ of the way from $100$ to $25$.
That means $$ e^{3k} = 1/3 $$ so $$ 3k = -\ln 3 \approx -1.1 $$ and $$ k \approx -1.1/3 \approx -0.37. $$
The not quite common knowledge is the value of $$ \ln 3 = 1.0986\ldots \approx 1.1 . $$ (More people know $\ln 2 \approx 0.7$, which leads to the "rule of $70$" for doubling time when compounding interest. This approximation tells you the "rule of $110$" for tripling.)