1) y = x2 - x-6
differentiate both sides with respect to X
dy/dx =( d(x2)/dx) - (d (x-6)/dx)
dy/dx. = 2x -(-6) (x-6-1 )
dy/dx = 2x +6x-7
2) y = (1+x2)4
dy/dx = (d(1+x2 )4 /dx) (d (1+x2 )/dx)
dy/dx = 4(1+x2 )4-1 (2x)
dy/dx = 8x (1+x2 )3
3) y = ln ( x3+x)
dy/dx = (d ( ln (x3 + x) / dx) (d (x3 + x)/dx)
dy/dx = (1/(x3 +x) ) (3x3-1 + 1)
dy/ dx = (3x2+1)/(x3+x)
1. Differentiate each of the following functions: (a) 22--6 (b) (1+) (c) In(x3 + x) (d)...
3. (6 points each) Differentiate the following functions. Do not simplify your answer. a. f(x) = 2*+1 In (x+1) b. p(x) = [arctan (x)]3 c. Q(x) = ex-sec(x)
Find the derivative of the following functions: Inx 22 C. a. F(x) = In (3x) b. Y=: Y=52x-1 d. Y= log1032 e. Y=In(x2-2)2/3 f. Y= In g. Y= log2 (2x - 1) h. Y= 8** e 1+e+
Compute the derivative (d/dx) of the following functions a. x3 - 2x b. (x+4)2 c. sin(ωx2) d. 3e-kx
5. Differentiate the following functions by filling in the chart. Original Function Rewrite Differentiate Simplify a. f(x)= V.x2 b. f(x)= c. f(x)= 4 3.x2 T d. f(x)= (5x)2
(d) f(x) = (1 + x) ln(1 + x) Hint: differentiate. (4) Expand the following functions into power series centered at 0 and find the radius of convergence. You can either use geometric series method, known expansions, or derivatives. You don't have to analyze the remainder.
9. Find the relative and absolute max and min of the following functions a. f(x)=x'+x b, f(x) = Vx+4 64 c. f(x) = x3-2x2 + 5 d. f(x)- x2+4 9. Find the relative and absolute max and min of the following functions a. f(x)=x'+x b, f(x) = Vx+4 64 c. f(x) = x3-2x2 + 5 d. f(x)- x2+4
(%) = u(x, y) + f 0(4,7) For each of the following functions, write as f(z) = u(x, y) + í v(x, y) and use the Cauchy-Riemann conditions to determine whether they are analytic (and if so, in what domain) a. f(z) = 2 + 1/(2+2) b. f(z) = Re z C. f(x) = e-iz d. f(z) = ez? 16 marks]
1. (6 marks) Provide the domain, target, and range of the following functions (a, b, c, d}3 . For each x E(a, b, cF, fx)-dx. a) fta, b. c}2 b) g: {a, b, c, d)-(a, b, c, d}?. For each x E(a, b, c, d], g(x) (4 marks) Use the ceiling and floor functions to qive a mathematical expression for the following a) Among a random group of 100 people at least 9 must be born in the same month...
The following exercises deal with the family of functions F(x) = x3 6. for λ > 0. a. Find all periodic points and classify them when 0< A<1. b. Prove that, if Ixl is sufficiently large, then lf"(x)| → oo. C. Prove that if λ is sufficiently large, then the set of points which do not tend to infinity is a Cantor set.
Find the derivative of the following functions: (x2-1) f(x) = (x2 +1) f(x) = (x3 + 2x)3(4x + 5)2