103.1.2.3.(D)The average rate of change is represented by the slope of secant segment . There appear to be 3 points at which the tangent lines are parallel to
.
4 Applications of Differential Calculus
(D) Substituting y = 2 yields x = 1. We find y′ implicitly.
3 y^2 y′ − (2xyy′ + y^2 ) = 0; (3y^2 − 2 xy)y′ − y^2 = 0.Replace x by 1 and y by 2; solve for y′.(A) 2 yy′ − (xy′ + y) − 3 = 0. Replace x by 0 and y by −1; solve for y′.(E) Find the slope of the curve at x = : y′ = x cos x + sin x. At x = , y′
= · 0 + 1. The equation is y − = 1 ( x − ).