University:
Georgia Southern University
Course:
MATH 1441 | Calculus I
Academic year:

2024
Views:

209

Pages:

3
Author:

Jalen W.

Solving an initial-value problem: An initial-value problem is one in which we are required to find a function that satisfies a) A differential equation and b) One or more initial conditions. Ex. 1: Find f if f'(x) = 3x^2 - 4x + 8, and f(1) = 9. f(x) = ∫f'(x) dx = ∫(3x^2 - 4x + 8) dx = x^3 - 2x^2 + 8x + C f(1) = 9 => 1 - 2 + 8 + C = 9 => C = 2 f(x) = x^3 - 2x^2 + 8x + 2 Ex. 2: Find f if f'(x) = 1 + 1/x^2 and f(1) = 2. f(x) = ∫f'(x) dx = ∫(1 + x^-2) dx = x - 1/x + C f(1) = 2 => 1 - 1 + C = 2 => C = 2 f(x) = x - 1/x + 2 Ex. 3: The velocity of a car (in feet per second) t sec after starting from rest is given by v(t) = 9t^2 - 3√t (0 ≤ t ≤ 30). Find the car's position, s(t), at any time t. Assume that s(1) = 8. v(t) = s'(t) s(t) = ∫s'(t) dt = ∫(9t^2 - 3t^(1/2)) dt = 3t^3 - 2t^(3/2) + C s(1) = 8 => 3 - 2 + C = 8 => C = 7 s(t) = 3t^3 - 2t^(3/2) + 7 Ex. 4: A company finds that the rate at which a seller's quantity supplied changes with respect to price is given by the marginal-supply function, S'(x)=0.24x^2+4x+10, where x is the price per unit, in dollars. Find the supply function if it known that the seller will sell 121 units of the product when the price is $5 per unit. The quantity supplied. S(x)=? if S^1(x)=0.24x^2+4x+10 X= unit price [S(x)=121 uh] when unit price x=5 [c=?] [121=\cdot08(5^{3})+2(5^{2})+10(5)] ⇒C=11 [S(x)=0.08x^{3}+2x^{2}+10x+11] [S(x)=\int S^{1}(x)dx] [\sigma(x)=\int(0.24x^{2}+4x+10)dx] [=0.24\frac{x^{3}}{3}+\frac{4x^{2}}{2}+10x+Q] [=0.08x^{3}+2x^{2}+10x+C] Ex. 5: A rock thrown upward with the initial velocity 50 ft/sec from 10 ft above the ground has a velocity modeled by (v(t)=-32t+50,) where t is the number of seconds after the rock is released and (v(t)) in feet per second. a) Determine a distance function h as a function of t (in this case, 'distance' is the height of the rock). b) Find the height and the velocity of the rock after 3 sec. If s(t) is the height function, then. [u(t)=S^{\prime}(t)=-32t+50] 10ft The height function s(t) = ((S'(t) dt) Find & using the initial condition that at t=0, s(t) = 10 ft [S(t)=-16t+50t+10] Ex. 6: Find the f line to the graph at any point ((x,f(x)) f^{\prime}(x)=3x^{2}-4x+1|) and the graph passes through the point (1, 1). (f(x)=\int f^{\prime}(x)dx) and (f(x)=1) at (x=1) [f(x)=x^{3}-2x^{2}+x+1] Ex. 7: Find f(x) if f'(x) = e^x + x, and f(0) = 2. f(x) = ∫f'(x)dx f(x) = ∫(e^x + x)dx f(x) = e^x + x²/2 + C Use the initial condition: at x = 0, f(x) = 2 2 = e^0 + 0 + C C = 1 ⇒ f(x) = e^x + x²/2 + 1 Ex. 8: An air conditioning company determines that the marginal cost, in dollars, for the xth air conditioner is given by, C'(x) = -0.2x + 500, C(0) = 0. Find the total cost of producing 100 air conditioners. C(x) = ∫C'(x)dx = ∫(-0.2x + 500)dx = -0.1x² + 500x + C Since C(x) = 0 at x = 0, C = 0 C(x) = -0.1x² + 500x

Initial Value Problems

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