Mixed Worksheet for Continuity and Differentiability Class 12

7. A company's profit function is given by

$$ P(x)=\left\{\begin{array}{ll} 10 x & \text { if } 0 \leq x \leq 100 \\ -0.1 x^2+40 x-2000 & \text { if } x>100 \end{array} .\right. $$

a. Check if the profit function is continuous at \(x=100\).
b. Check if the profit function is differentiable at \(x=100\).
c. What do your answers to (a) and (b) signify for the company's profit model?
Does the rate of change of profit change smoothly at the 100 -unit mark?

If \(\mathrm{x}=\mathrm{e}^{\mathrm{t}+\frac{1}{\mathrm{t}}}\) and \(\mathrm{y}=\mathrm{e}^{\mathrm{t}-\frac{1}{\mathrm{t}}}\), then find \(\frac{\mathrm{dy}}{\mathrm{dx}}\) at \(\mathrm{t}=-2\).