Which of the following gives the equations of both lines through the point $\left(2,,,,-,3\right)$ that are tangent to the parabola $y=x^2+x$?

Checking the Mean Value Theorem

Which of the functions in Exercises 7–12 satisfy the hypotheses of the Mean Value Theorem on the given interval, and which do not? Give reasons for your answers.

f(x) = {x² − x, −2 ≤ x ≤−1

2x² − 3x − 3, −1 < x ≤ 0

Given the equation $e^x-y=x{y}^3+e^2-18$, what is the value of $\frac{dy}{dx}$ at the point $(2,2)$?

Given the equation $x+3y{y}^{13}$=$y$, what is $\frac{dy}{dx}$ at the point $(2,8)$?

Given $x=e^t$ and $y=t{e}^{-t}$, which of the following is the correct value of $\frac{dy}{dx}$ at $t=1$?

118. A particle is traveling upward and to the right along the curve y=ln(x). Its x-coordinate is increasing at the rate (dx/dt)=√x m/sec. At what rate is the y-coordinate changing at the point (e², 2)?

A sphere is growing at a rate of $50\(\frac{\operatorname{\mathrm{cm}\)}^3}{s}$. At what rate is the radius of the sphere increasing when the radius is $5\(\operatorname{\mathrm{cm}\)}$?

A right tringle has a base of and a height of . The height of the right triangle is decreasing at a rate of , at what rate is the area of the triangle decreasing?

The perimeter of a rectangle is fixed at . If the length $L$ is increasing at a rate of , for what value of $L$ does the area start to decrease? Hint: the rectangle's area starts to decrease when the rate of change for the area is less than 0.