20. La derivada \( \frac{d x}{d y} d e Y=(x+1)(x-1) \) es: a. \( 3 X \) b. \( X-1 \quad y=x^{2}-1 \) c. \( X+1 \) \( 2 X \)

Next, find the horizontal asymptote, If any. This is done by determining the ratio of the degree of the polynomial in the numerator to the degree of the denominator polynomial. \[ g(x)=\frac{6 x}{x^{2}+5 x-6} \] degree of numerator degree of denominator If the ratio is less than one, then \( y=0 \) is a horizontal asymptote. If the ratio is equal to one, then the horizontal asymptote is determined by the quotient of the leading coefficients of the numerator and denominator polynomials. If the ratio is greater than one, the graph of \( g \) has no horizontal asymptote. The ratio of the degree of the numerator to the degree of the denominator is - Select- 1 . Therefore, the graph has the following horizontal asymptote(s). (Enter your answers as a comma- separated list of equations. If there are no asymptotes, enter DNE.) Submit Skin(youcannolcome back)

Set up the integral that gives the volume of the solid. Use increasing limits of integration. Select the correct choice below and fill in the answer boxes to complete your choice. (Type exact answers.) A. \( \int(2 \pi x) f\left(x^{2}\right) d x \) B. \( \left.\int^{2}\right)^{2} \mathrm{dx} \) The volume of the solid is Type an exact answer.)

3. [5 points] Evaluate by interpreting in terms of areas. \( \quad \int_{-2}^{5}(5-|x|) d x \)

Q1. Find an extremal for the functional \[ f(x)=\int_{1}^{t_{f}}\left[2 x(t)+\frac{1}{2} \dot{x}^{2}(t)\right] d t \] The boundary conditions are \( x(1)=4, x\left(t_{f}\right)=4 \) and \( t_{f}>1 \) is free.

The rate at which a machine operator's efficiency, \( E \) (expressed as a percentage), changes with respect to time \( t \) is given by \( \frac{d E}{d t}=35-12 t \), where \( t \) is the number of hours the operator has been at work. a) Find \( E(t) \), given that the operator's efficiency after working 2 hr is \( 79 \% \); that is, \( E(2)=79 \). b) Use the answer in part (a) to find the operator's efficiency after 3 hr ; after 5 hr .