- B We can use Implicit Differentiation to find . First, differentiate with respect to x: 3 y^2 = x
+ y − 4x. Next, plug in (1, 2) for x and y: 3(2)^2 = (1) + (2) − 4(1). Simplify: 12 = − 2. And solve for on the left and the terms without on the right: = −.- D If we take the limit as x goes to 0, we get an indeterminate form , so let’s use L’Hôpital’s
Rule. We take the derivative of the numerator and the denominator and we get = . Now, when we take the limit we get .
16. A We can evaluate this integral using u-substitution. Let u = 1 + x^2 and du = 2xdx, so du = xdx.
Substitute into the integrand: ∫ x sec^2 (1 + x^2 )dx = (^) ∫ sec^2 u du. Integrate: ∫ sec^2 u du =
tan u + C and substitute back: tan u + C = tan(1 + x^2 ) + C.
- B First, we need to find . It’s simplest to find it implicitly.
18 x + 32y = 0Now, solve for .Next, plug in x = 2 and y = −1 to get the slope of the tangent line at the point.Now, use the point-slope formula to find the equation of the tangent line.(y + 1) = (x − 2)