When you have a function for example of the form s = f (t) where "s" is the distance and this depends on the time it is said that its derivative expresses the velocity of the particle at the "t" units of time, this velocity concept corresponds to the instantaneous variation rate of "s" per unit variation of "t". More generally, if a quantity "y" is a function of an amount "x" in the form y = f (x), the variation of "y" with respect to "x" can be expressed in the following way: Δy / Δx, which is the same as f '(x) if it exists.
Let V (x) cubic centimeters be the volume of a cube of edges x cm. Calculate the instantaneous rate of variation of its volume when x = 5. The volume of a cube is x3 and V'(x) = 3x2 then V'(5) = 3(5)2 : 75 cm. Then the instantaneous rate of variation of V (x) with respect to x when x = 5 is V '(5) = 75 cm. We have a circle of radius r, calculate the instantaneous rate of variation of its area when r = 10. The area of a circle is: Ac = π.r2 when calculating Ac '(r) = 2.π.r and Ac' ( 10) = 2.π.10 = 62.8 cm.
In economics the concept of marginal variation which is the rate of instantaneous variation of one quantity with respect to another is widely used. If C (x) represents the total cost of producing x quantity of items then C '(x) is the marginal cost function, this being its variation rate with respect to "x". Let's see some exercises:Suppose that C (x) dollars is the cost of manufacturing x items. Calculate the marginal cost function and its variation rate when x = 5. C(x) = 0.55x2 + 0.04x The marginal cost function is C '(x): 1.1x + 0.04 and
C '(5) = 5.54 dollars. Then when 5 items are manufactured the price varies 5.54 dollars.We also have the function total income that is denoted by R and is defined by R (x) = px. Where R (x) dollars is the income received when selling x units to p dollars per unit. The derivative of R (x) is the marginal income when x units are sold and this variation rate can be positive,
negative or zero. Let's see an example and several exercises solved. Suppose that R (x) dollars is the total income from the sale of x products and R(x) = 350x – 0.15x3 Determine the marginal income function and the marginal income when x = 35. The marginal revenue function is R'(x) 350 – 4.5x2 and the marginal revenue is R' (35) = 350 - 183.75 = 166.25 dollars. Then the variation rate when 35 products are sold is $ 166.25 per product.
Let m (x) be the slope of the line tangent to the curve y = 3x3 + 2x2 + 5 at the point (x, y), determine the instantaneous rate of change of m (x) with respect to x at point (1, 10). It is by definition that m (x) = dy / dx
= 9x2 + 4x. The instantaneous rate of variation of m (x) with respect to x is determined by m '(x) or d2y/dx2 m' (x) = 9x + 4 at point (1, 10) m '(1) = 9 ( 1) + 4 = 13
Let's see several exercises to synthesize the above.
1) Let A(x) cm2 be the area of a square of side x cm, determine the rate of change of A (x) when x = 5. We have that the area of a square is equal to x2, the rate of variation is A '(x) = 2x which is evaluated at 5 = 10 cm2.
2) Stefan's law for the emission of radiant energy "R" states that R(T) = kT4,, determine R '(T) when T = 200 Kelvin. Since "k" is a constant derivative, we have that R' (T) = 4.(kT)3 Then R' (T) = 4.(k.200)3
3) A straight circular cylinder has a constant height of 10.00 inches. Let V inch3 be the volume of the cylinder and r inches the radius of your base. Determine V '(r) when r = 5.00. The volume of a cylinder is equal to its base by height then V(r) = (π.r2)H then the rate of instantaneous variation of the solid with respect to the radius is equal to V '(r) = 2 (π.r) H that evaluated in r = 5 is: V '(5) = 2 (π.5) H = (31.4) H ,4) We have a pyramid whose base is equal to the cube of its height. Calculate the instantaneous variation rate of volume with respect to height. It is requested to calculate V '(h). The volume of a pyramid is V = 1 / 3bh. But b = h3, then V = 1/3(h3h) = 1/3(2h3). Derivating with respect to h we have: V'(h) = 1/3(6h2).
5) Let r be the radius of a straight circular metal plate of area A (r) square inches and circumference of C (r) inches. If the heat expands the dish determine the instantaneous rate of variation of A (r) and C (r) with respect to r. The area of a circle is Ac = π.r2 and the length of a circle is Lc = 2rπ then A'c = 2π.r and L'c = 2π so it is concluded that the instantaneous variation rate of the length of a circumference is a constant with respect to radius . 6) A solid consists of a straight circular cylinder and a hemisphere at each end, the length of the cylinder is twice its radius. Let r units be the radius of the cylinder and the hemispheres and V (r) units cubicize the volume of the solid. Calculate V '(r). The volume of a cylinder is equal to its base by height then V(r) = π.r2.H but H = 2r then V(r)= π.r2.2r = π.3r2 and the volume of a sphere is 4/3. π.r3,, then the volume of the solid is equal to V(r) = (π.3r2) + 4/3(π.r3) V'(r) = 6(π.r) + 12/3(π.r2).
7) Boyle's Law for the expansion of a gas establishes that PV = C, where P is the pressure, V the volume and C a constant. Determine V '(P) when the pressure is 4. For this calculation we must express V in function P, then V = C / P. Differentiating with respect to P we have: 1) V'(P) = -(C/P2), then: V'(4) = -(C/42). 8) A bacterium has a spherical shape, determine the instantaneous variation rate of the surface area of the bacteria with respect to the radius. The surface of a sphere is S(r) = 4.π.r2 and S'(r) = 8.(π.r) 9) Sand is poured into a conical mound so that its height is twice its radius. If V is your volume, calculate the derivative of V (r) at r = 5. The area under study is a cone and the volume of a cone is V(r) = 1/3(π.r2)H. But H = 2r then V(r) = 1/3(π.3r2) and V '(r) = 6/3 (π.r) so V' (5) = 6/3 (π.5) = 31.4 cubic units. 10) The number of dollars of the total cost of manufacturing x watches in a certain factory is given by C(x) = 1500 + 3x + x2. Determine the marginal cost function and the marginal cost when x = 40. The marginal cost function is C '(x) = 3 + 2x and C' (40) = 3 + 2 (40) = 83 dollars.
11) The total income received from the sale of x desks is R (x) dollars, where R(x) = 200x – 1/3x2. Determine the marginal revenue function and the marginal revenue when x = 30. The marginal cost function is R '(x) = 200 - 2 / 3x and R' (30) = 200 - 2/3 (30) = 180. 12) If R (x) dollars is the marginal revenue received from the sale of x television sets, where R(x) = 600x – 1/20x3, determine the function and marginal revenue when x = 20. The income function marginal is: R'(x) = 600 – 3/20x2 and R'(20) = 600 – 3/20(20)2 = 540 dólares. 13) If C (x) dollars is the total cost to manufacture x paperweight, and C(x) = 200 + 50/x + x2/5 .Determine the function and marginal cost when x = 10. The marginal cost function is 1) C'(x) = -(50/x2) + 2x/5 and C'(10) = -(50/x2) + 2x/5 = -0.5 + 4 = 4.5 dólares.
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