MAT 1332: CALCULUS FOR LIFE SCIENCES

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1 MAT 1332: CALCULUS FOR LIFE SCIENCES JING LI Contents 1 Review: Functions of several variables I: Partial derivatives 1 2 Function of several variables II: Vector-valued functions 2 21 Introductory example 2 22 Definition 2 23 Linear approximation and the Jacobian matrix 4 1 Review: Functions of several variables I: Partial derivatives Definition of partial derivative of functions of two independent variables Suppose that f is a function of two independent variables x and y The partial derivative of f with respect to x is defined by f(x, y) x = lim h 0 f(x + h, y) f(x, y) h The partial derivative of f with respect to y is defined by f(x, y) y = lim h 0 f(x, y + h) f(x, y) h Tangent plane Let f(x, y) be a real-valued function of two variables If the tangent plane to the graph of f at the point (x 0, y 0, z 0 ) = (x 0, y 0, f(x 0, y 0 )) exists, then it is given by the equation z z 0 = f(x 0, y 0 ) x (x x 0 ) + f(x 0, y 0 ) (y y 0 ) y The linear approximation The linear approximation of a function f(x, y) near a point (x 0, y 0 ) is given by L(x, y) = f(x 0, y 0 ) + f(x 0, y 0 ) (x x 0 ) + f(x 0, y 0 ) (y y 0 ) x y [ [ f(x0, y 0 ) f(x 0, y 0 ) x x0 = f(x 0, y 0 ) +, x y y y 0 provided the function is differentiable Date:

2 2 JING LI 2 Function of several variables II: Vector-valued functions 21 Introductory example So far, we have only considered real-valued functions, f : R n R (x 1, x 2,, x n ) f(x 1, x 2,, x n ) Now, we want to be able to describe several quantities that are all dependent on the same variables Recalling what we have learned in Section 55: Two-Dimensional Differential Equations, for example, we described a predator-prey system as db = (λ ɛp)b = λb ɛpb = f 1 (b, p) dp = ( δ + ηb)p = δp + ηbp = f 2 (b, p) where b(t): the population of bacterial at time t ( prey ); p(t): the population of amoebas at time t ( predation ); λ: per capita production rate of the bacterial; ɛp: the rate at which an individual bacterium is eaten; δ: negative per capita rate of amoebas in the absence of their prey; ηb: the rate at which an amoeba eats bacteria Hence, the growth rate of the prey and the predator both depend on the densities of prey and predator We can write the right-hand side of the above system as a single, vector-valued function of the two variables b, p as follows: F (b, p) = [ f1 (b, p) f 2 (b, p) = [ λb ɛpb δp + ηbp 22 Definition The above example is one of the examples of vector-valued functions More generally, the vector-valued function will be defined as follows: Definition A vector-valued function F of the variables x 1,, x n is a function of the form ie, F : R n R m (x 1, x 2,, x n ) F (x 1, x 2,, x n ) = f 1 (x 1, x 2, x n ) f m (x 1, x 2, x n ) f 1 (x 1, x 2, x n ) f m (x 1, x 2, x n ) where, each function f i (x 1, x 2,, x n ) is a real-valued function, f i : R n R (x 1, x 2,, x n ) f i (x 1, x 2,, x n ) Example 1 Dynamics of Competition da db where ( a + b) = r a 1 a K a = r b ( 1 a + b K b ) b

3 MAT 1332: CALCULUS FOR LIFE SCIENCES 3 a(t): the population of bacteria type a at time t; b(t): the population of bacteria type b at time t; r a : the maximum capita production rate for type a; r b : the maximum capita production rate for type b; K a : the carrying capacity; K b : the carrying capacity Example 2 Newton s Law of Cooling dh da where H(t): the object temperature at time t; A(t): the ambient temperature at time t; α: constant of proportionality; α 2 : constant of proportionality = α(a H) = α 2 (H A) Example 3 Evaluate the function at points (1, 0), (1, π 2 ), (2, π) (1) F (1, 0) = [ 2x 2 y 3y + x e x sin y (2) F (1, π 2 ) = (3) F (2, π) =

4 4 JING LI 23 Linear approximation and the Jacobian matrix Our main task in this subsection will be to define the linear approximation (linearization) of vector-valued functions where the domain and the range are R 2 That is, we only consider the case of two variables and two equations, or equivalently, vector-valued functions with two components, ie, [ f(x, y) g(x, y) From the previous section, we know how to find the linear approximation for each of the two functions, f(x, y) and g(x, y), namely [ x x0 for function f(x, y), f(x, y) f(x 0, y 0 ) + grad(f) y y 0 [ x x0 for function g(x, y), g(x, y) g(x 0, y 0 ) + grad(g) y y 0 Now we put these two linear approximation together to obtain the linear approximation for F (x, y): Definition The Jacobian Matrix & Linear Approximation The matrix of partial derivatives [ [ f(x,y) f(x,y) grad(f) x y J(x, y) = = g(x,y) g(x,y) grad(g) x y [ f(x, y) is called the Jacobian Matrix of the function g(x, y) Using the Jacobian matrix, we can write the linear approximation to F (x, y) as [ x x0 L(x, y) = F (x 0, y 0 ) + J(x, y) (x0,y 0 ) y y 0

5 MAT 1332: CALCULUS FOR LIFE SCIENCES 5 Example 4 Find a linear approximation to [ (x y) 2 2x 2 y at point (2, 3) Use your result to find an approximation for F (19, 31) and compare the approximation to the value of F (19, 31) when you use a calculator (1) F (2, 3) = (2) To find the Jacobian matrix, (3) To find the linear approximation L(x, y), (4) To compare the values: Using the linear approximation, F (19, 31) L(19, 31) = Using calculator, F (19, 31) = Example 5 Find a linear approximation to [ 2x 2 y 3y + x e y cos x at point (0, 0) Use your result to find an approximation for F (01, 01) and compare the approximation to the value of F (01, 01) when you use a calculator (1) F (0, 0) =

6 6 JING LI (2) To find the Jacobian matrix, (3) To find the linear approximation L(x, y), (4) To compare the values: Using the linear approximation, F (01, 01) L(01, 01) = Using calculator, F (01, 01) = Example 6 Find a linear approximation to [ 2x + y x y 2 at point (1, 2) Use your result to find an approximation for F (105, 205) and compare the approximation to the value of F (105, 205) when you use a calculator (1) F (1, 2) = (2) To find the Jacobian matrix, (3) To find the linear approximation L(x, y),

7 MAT 1332: CALCULUS FOR LIFE SCIENCES 7 (4) To compare the values: Using the linear approximation, F (105, 205) L(105, 205) = Using calculator, F (105, 205) = Remark: Some examples in this note are cited from the book titled Calculus for Biology and Medicine by Claudia Neuhauser