THEOREM 11.5.2 THE CAUCHY MEAN-VALUE THEOREM* Suppose that f and g are differentiable on (a, b) and continuous on [a, b]. If g is never 0 in (a, b), then there is a number rin (a, b) for which f'(r) f(b) - f(a) g'(r) = g(b)g(a)
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Explain the key points of 11.5.2
![THEOREM 11.5.2 THE CAUCHY MEAN-VALUE THEOREM*
Suppose that f and g are differentiable on (a, b) and continuous on [a, b]. If
g is never 0 in (a, b), then there is a number rin (a, b) for which
f'(r) f(b) - f(a)
g'(r)
=
g(b)g(a)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2b47420a-cb3e-467a-b067-9fd1f65b8aee%2Ff7f01f09-c7ab-4386-a925-8f09e73b0f62%2Ffedd4hb_processed.jpeg&w=3840&q=75)
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- 10) The function f below satisfies Rolle's Theorem in the interval [a, b] because (a) f is continuous on [a, b] (b) f is differentiable on (a, b) (c) f(a) = f(b) = 0 (d) All three statement; A, B, and C (e) None of theseLet f and h be real-valued functions continuous on [a, b], differentiable on (a, b), and h(a) not equal h(b). Prove c exists in (a, b) so that (f(b)-f(a))h'c=f'(c)(h(b)-h(a))Find continuous functions f1, f2, f3, f4 defined on the open unit interval I = (0, 1) (i.e. functions fi : (0, 1) → R) such that f1(I) = (L1, M1) f2(I) = (L2, M2] f3(I) = [L3, M3) f4(I) = [L4, M4]for some real numbers Li, Mi.
- According to the Mean Value Theorem, if f (x) is a function that is continuous on a, b , and is differentiable on (a, b), then there exists at least one point CE (a, b) such that f' (c) is equal to the slope of the secant line passing through the points (a, f (a)) and (b, f (b)) . Consider the function f (t) 2t3 6t? + 8 on the interval [-1, 2]. Find the = - - set of values of c for which such that f' (c) is equal to the slope of the secant line passing through the points (-1, f (–1)) and (2, f (2)) . Use set roster notation and round the elements to the nearest hundredth. Write elements in ascending order.Plzb) Euler's theorem states that f(') = "lax, X1 + af lara x2 for functions that are homogenous of degree 1. Show that the function below is homogenous of degree 1 and that it obeys Euler's theorem f(x,x2) = x} + x3
- Determine L[f]if 0, f(1) = {1- 1, 0 2. 1,4. Let [a, b] be an interval in R and let be a contimuous function on [a, b). (i) Check that the function r+ (b- a)x + a is a bijection from [0, 1] to [a, b] and its inverse is I+E. (ii) For r € [0, 1], let g(x) = »((b-a)x+a) (which is equivalent to say that (r) = g(). Check that g is continuous on [0, 1]. (iii) By question 3), we know that there exists a sequence of polynomials, say (4n)n, which converges uniformly to g on [0, 1]. Construct a sequence of polynomials (pn)n such that || – Pn|.ja.bj = ||g – (iv) Deduce the Weiestrass approximation theorem for the continuous function on [a, b].2. a. f(x) = (x+2)/(x +4). Use the Mean Value Theorem on the interval [0, 4] to find all values of x in the interval (0, 4) such that the value of the first derivative of f at those points is (f(b) – f(a))/(b – a). b. For the interval [-5, 0], fb) - fa))(b — а)-GG0) — f-5)(0 — (-5)) = (½ – 3)/(0 + 5) = -½ but, since the first derivative of f(x) is always positive, nowhere on that interval does the first derivative of f(x) = -½. Thus the Mean Value Theorem does not hold for f(x) on this interval. Why?
- Theorem 1 (Rolle's Theorem) Suppose f is a continuous function in the domain (a, b). If f(a) = f(b) = 0, then a number c exist in (a, b) such that f'(c) = 0. This theorem is illustrated in Fig. 1.1. Exercise 1.1: For the function f(x) = sin(x) in the domain (0,7), (a) Sketch the function f(x) in the domain specified above. (b) What is the value of f(0) and f(T). (c) What is the value of c such that f(c) = 0 ? Your answer should confirm that 06. Use the Mean Value Theorem to find all values of c in (6, 14) such that ƒ'(c) = f(b)-f(a) b-a for f(x)=√√√2x - 3 on the interval [6, 14]
