Consider the function f(x) = {(log(x - 1))/2, if |x - 10| > 10^-10; x + 1, other
ID: 3123048 • Letter: C
Question
Consider the function f(x) = {(log(x - 1))/2, if |x - 10| > 10^-10; x + 1, otherwise. (a) Sketch the graph of f in the interval [2, 15]. (b) Compute, if possible, the limit of f when x goes to 10. Compute the derivative of the following functions: (a) f(x) = log_2(x^2 - x) + (x^4 - 1)^12. (b) g(x) = 3(x^3 - 1) - 2x/x^5 - 3x^4. f: R rightarrow R is derivable on x = -1.7, point in which it attains a local minimum value of 3. Determine the equation of the tangent line to the graph of f at the point corresponding to x = -1.7. Consider the following function f: R^2 rightarrow R given by f(x, y) = 4x^2 + 3y^2 - xy + 2x - 1. (a) Compute the gradient of f. (b) Find the critical points of f. (c) Graph the level curve C of f that passes across the point (1, 1). (d) Parametrize C by specifying a function alpha: I rightarrow C, where I is interval of R. (e) Find a value t Element l such that alpha(t) = (1, 1). (f) Show that the gradient of f in the point (1, 1) is orthogonal to C at this point. Verify whether the given function y = f(x) is a solution of the specified equation: (a) f(x) = sin(x); d^2y/dx^2 + y = 0. (b) f(x) = x^2 + pix; xdy/dx - x^2 - y = 0. (c) f(x) = cos (x^2 + ln(x)); (2x + 1/x) y" - (2 - 1/x^2)y' + (2x + 1/x)^3 y = 0. (d) f(x) = x^2 arctan(x); y"' - 2y" + y = 1.Explanation / Answer
10 .) a.
f(x) =y =sinx
dy/dx = cosx
d2y/dx2 = -sinx
so , d2y/dx2 + y =0
-sinx + sinx = 0
b.) f(x) = y =x*x + x
dy/dx = 2x +
so ,
xdy/dx - x*x -y =0
2x*x + x - 2*x*x -x = 0
c.)
f(x) =y = cos(x*x + ln(x) )
y'= dy/dx = -sin(x*x + ln(x) ).( 2x + 1/x )
y'' = -cos(x*x + ln(x)).(2x + 1/x )(2x + 1/x ) -sin(x*x + ln(x) ).(2- 1/x*x)
7.)
a.)
f(x) = log(x*x-x) + (x^4 -1 ) ^ 12
= 2x-1/x(x-1) + [(x^4 -1)^11 ] * 3.x^4 / 2
b.)
=3 ^ (x^3 -1 ) - 2x/x^5 -3.x^4
= 3 ^ (x^3 -1 ) - 2/x^4 -3.x^3
= 3 ^ (x^3 -1 )log3.(3x*x) - [ (x^4 -3x^3)d/dx(2) - 2.d/dx(x^4 - 3x^3) ]/( x^4-3x^3)^2
= 3 ^ (x^3 -1 )log3.(3x*x) - 2(3x^4 - 9x^2) / (x^4 -3X^3)^2
9.) a.)
dy/dx = 8x + 6ydy/dx - [y + xdy/dx ] + 2
dy/dx = 8x -y +2 / 1+x -6y