Recall that for a function f: R Rk, and a point To R", the total derivative of at o is a linear function from Rn to Rk that satisfies a limit formula which we covered in lecture. It is written as D|, The matrix of Dfis therefore a k x n matrix; its components are all of the partial derivatives of f (in other words, the Jacobian matrix): 21, (3) = To The defining feature of the total derivative D is that, if you make a small change to the inputs of f, say Ar, then the total derivative tells you approximately how much the output of f will change by: D |A Ax This is the multivariable linear approximation formula. 2. The flow rate of blood through a blood vessel depends on the inner radius (r) of the blood vessel (in mm), as well as the pressure (p, in mmHg) and temperature (7, in C) of the blood in the blood vessel. This flow rate B has been modeled by the following function: 1 mL r(p 40) (1 e0.05T) (in 170 S Suppose that in a certain individual, under normal conditions, one particular artery has a radius of 2 mm, and the pressure in that artery is 105 mmHg and the temper- ature of the blood is 37C. Suppose the person lights up a cigarette, and runs up a flight of stairs. Nicotine is a vasoconstrictor, so the radius of the artery decreases by 0.3 mm. And the sudden exercise of running both increases the pressure in the artery by 1.2 mmHg, and increases the temperature by 0.1 C. By approximately how much does the flow rate of the blood through the artery change as a result of this? B(r, p, T) = =