1. The supply equation for a certain kind of native bags is x = 3 +1/4 p² Where p pesos is the price per bag when x units are supplied

a. Find the rate of change of the supply when the price is

increased from P40 to P42?

b. Find the instantaneous rate of change of the supply with

respect to the price when the price is P40 Hint: x¹=f(x+h)-f(x) / h = f(40+2)-f(2) / ? 2. For a certain manufacturer the revenue R obtained from the sales

of x units of his product is given by R = 50 x - 0.3 x 2. Find the

instantaneous rate of change of R with respect to x. How fast

does R change with respect to x when x =5

H =d+1h ll the time elapsed oil! is very smell .we would expect the average velocity lo be very close to the instantaneous velocity at t= 2. thus we can compute the instantaneous velocity Iii-n, by the limit "in: = ELI; \"turd =HIGIHSM = 5'4 Thereiore seam seconds .1he telling objectis eevelling at the rate of 34 test per second. The derivative 0'! a function : The derivative of the function x} with respect to x is the function f'ix] Read as (f prime nix) given by: f(x) = L13; fearful and the process of computing the derivative is called the differentiation we say the! ma is differentiable at c it f'f e ) exists ( that is the limit at the difference quotient exist when x = c.) Example 1. Find the derivative of f{x) = :9 Solution: Using the formula above {surf-:5 n m as 3 i+ *1 *3_3 =Iimg'm 3\" + 1' =l|i_.rd.(3x=+3xh+h2] hm n = 3x! Example 2 .A manufacturer estimates that when x units of a certain commodity are produced and sold .the revenue derive will be Rut] = 0.51:1 + 3.x -2 thousand pesos .At what rate is the revenue changing with respect to the level of production it when3 units are being produced? Ie the revenue increasing or decreasing at this time? Solution: Since it represents the number of units produced .we must have x 2 0 The difference quotient of His] is singem [mstx3+zxn+n3J-r 3tx+n}2)]-v[o5x3+3x2] it ' i. a 3%: x+0.5h+3 Thus the derivative of R00 is apt-inJ-sm R' x = lim ( i M h mb+05h+31=x+3 Andsfncex=3 R'(3} =(3i+3 =6 It foilcws that the revenue is changing at the rate ofP 6,000 per unit with respect to the level of production when 3 units are being produced. Applicationn 1' the Supply enuation far a certain kind of native cage is x a 3 {-i p3, Where p oases is the price per bag when x units are supplied a. Find the rate of chance of the suonl'ir when the Drice is increased _. .l a