Download
1 / 6
70 likes | 239 Vues
This guide explores the application of the Mean Value Theorem (MVT) to the function f(x) = sin(x) over the interval [1, 1.5]. Following the theorem's formula, we derive that the difference in function values, f(1.5) - f(1), equates to the function's derivative at some point c in the interval. By setting up the equation and solving for c, we find that c = 1.253, thereby demonstrating that this value satisfies the conditions of the MVT within the specified interval.
E N D
Jarrod Asuncion Period 1 Brose Mean value theorem
Equation f(b) – f(a) = f’(c) b – a Slope = f’(c)
Sample Problem Find the number c satisfying the Mean Value Theorem for f(x)=sinx on the interval [1,1.5], correct to three decimal places.
Step 1: Use formula [1,1.5] is the same as [a,b]. The function is f(x) = sinx. f(1.5) – f(1) = 0.997 – 0.841 = 0.312 = f’(c) 1.5 – 1 0.5
Step 2: Take derivative of function and set it equal to f’(c) and substitute ‘x’ for ‘c’. And solve for ‘c’. f(x)=sinx =› f’(x)=cosx f’(x)=f’(c) & substitute ‘x’ for c. trying to find ‘c’ cosc=0.312 c=1.253
IMPORTANT The value for ‘c’ must be between the interval given. c=1.253 Interval [1,1.5] C satisfies the Mean Value Theorem (MVT)
