= Solution
Differentiating the supplied relation gives
$$
\frac{d\rho}{d\omega}=\Lambda+\frac{\beta\lambda}{(\omega+\lambda)^2},
\qquad\left.\frac{d\rho}{d\omega}\right|_0=\Lambda+\frac\beta\lambda.
$$
Thus the tangent approximation is $\rho\simeq\omega(\Lambda+\beta/\lambda)$ and
$$
\boxed{T_{\rm tangent}=\frac{\Lambda+\beta/\lambda}{\rho}=70.5\ \mathrm s.}
$$
It is about $16.4\%$ longer than $60.58\ \mathrm s$ and therefore predicts slower growth. For this positive step it is not conservative as a growth-rate estimate. The local expansion needs $|\omega|\ll\lambda$; here the actual ratio is approximately $0.165$, so its error is appreciable.
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