(a) Use
(b) If f and g are inverse functions and f′ is continuous, prove that
[Hint: Use part (a) and make the substitution y = f(x).]
(c) In the case where f and g are positive functions and b > a > 0, draw a diagram to give a geometric interpretation of part (b).
(d) Use part (b) to evaluate
Trending nowThis is a popular solution!
Chapter 7 Solutions
Calculus: Early Transcendentals
- Integral of the constant function f(x) = k is: C 0 kx+C k+Carrow_forwardOn the set of continuous functions in the range [−1,1] Which two functions below are perpendicular to each other according to the inner product defined as ⟨f,g⟩=∫1−1f(x)g(x)dx?arrow_forwardUse the Intermediate Value Theorem to show that f(x) = cos^−1 (x) − e^x has a zero in the interval [0, 1]arrow_forward
- Find constants c1 and c2 such that F(x) = c1x sin x + c2 cos x is an antiderivative of f (x) = x cos x.arrow_forwardF(x) = x t(t4 + 1) dt 0 (a) Integrate to find F as a function of x. (b) Demonstrate the Second Fundamental Theorem of Calculus by differentiating the result in part (a).arrow_forwardWhich of the following is FALSE? A. integrate f(x) dx from b to a = - integrate f(x) dx from a to b B. integrate kf(x) dx from a to b = k * integrate f(x) dx from a to b c. integrate [f(x) + g(x)] dx from a to b = integrate f(x) dx from a to b + integrate g(x) dx from a to b D. integrate [f(x) * g(x)] dx from a to b = integrate f(x) dx from a to b * integrate g(x) dx from a to barrow_forward
- true or false with reason c) If f and g are continuous and f(x) ≥ g(x) for a ≤ x ≤ b, then integrationbaf(x)dx>integrationbag(x)dxarrow_forward1. Use the definition of the derivative of complex function to show that f(z) is not differentiableat z = 0 wheref(z) = conjugate of z3/ z2 , if z is not equal to 0 and f(z) = 0, if z =0arrow_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage