please explain your solution with details. 7) Let f and g be differentiable functions such that...
F. Te QB 2. (10 points) Let f and g be differentiable functions of x, and c an arbitrary positive constant number. Find the derivative of the function tanx h(x) - Vc + f(x) sec x Use the f' and g' notation. Size 40.72 KB g(x) Well
(a) Can there be differentiable functions f,g (on R) with g(0)-f(0) 0 and f()g(x) for all z E R? What about if we ask (only) for continuous functions f,g? (a) Can there be differentiable functions f,g (on R) with g(0)-f(0) 0 and f()g(x) for all z E R? What about if we ask (only) for continuous functions f,g?
21 Let f and g be functions from R3 to R. Suppose fis differentiable and V f(x) - g(x)x. Show that spheres centered at the origin are contained in the level sets for f; that is, f is constant on such spheres.
4) IVAN TUJU.7. LT If y = f(x) and y = g(x) are differentiable functions with values of the functions and their derivatives as indicated in the table below, compute the derivative (5(8(x)) + g(f(x))) evaluated at x = 4: * f(x) f(x) g(x) g(x) 4 5 8 8 11 5 4 12 8 1124 10
1. Let f 1 , f 2 , … , f n be differentiable functions. Prove, using induction, that ( f 1 + f 2 + ⋯ + f n ) ′ = f ′ 1 + f ′ 2 + ⋯ + f ′ n You may assume ( f + g ) ′ = f ′ + g ′ for any differentiable functions f and g . 2 .Make up a sequences that have : 1, 2, 4,...
Real analysis 7. Assume that f and g are differentiable functions such that f(0) 9(0) and that for all & ER, S' () > '(x). Prove that f(c) > 9() for all > 0.
6. Suppose that f and g are differentiable functions on an interval (a, b), and suppose that for all (a, b) we have f'(z) = g(z) and g'(z) = -f(x). Show that f2+g2 is constant.
MATH119 - Final tion 6 Let f and gbe differentiable functions on R and consider the function et answered h(z) = 5" }()g()dt ed out of 5 ag question Which of the following is equal to h'()? (Warning. Be careful that, in the definition of h(x) the definite integral is with respect to t.) Select one: a. f(2).9(3) A b. [*}"(w)g(t)dt + f(z)g(0) c. f'(x) · g(x) d. [ 7)g(e)dt + f(2)() eſ sz)ge)dx + $()() Next page
Let f and g be differentiable on R such that f(1) = g(1), and f'(x) < '() for all r ER. Prove that f(x) = g(2) for 3 >1.
(8) Let E c R" and G C Rm be open. Suppose that f E -G and g:GR', so that h -gof:E R'. Prove that if f is differentiable at a point x E E and if g is differentiable at f(x) є G, then the partial derivatives Dh,(x) exist, for all , SO , . . . , n, and and J-: に1 The subscripts hi, 9i, k denote the coordinates of the functions h, g, f relative to...