91Ó°ÊÓ

Evaluate \(\lim _{x \rightarrow 1} \frac{x^{1000}-1}{x-1}\)

Find \(d y / d x\) by implicit differentiation. \(e^{y} \cos x=1+\sin (x y)\)

Find \(d y / d x\) by implicit differentiation. \(\sin x+\cos y=\sin x \cos y\)

Draw a diagram showing two perpendicular lines that intersect on the \(y\) -axis and are both tangent to the parabola \(y=x^{2} .\) Where do these lines intersect?

\(77-80\) Use implicit differentiation to find an equation of the tangent line to the curve at the given point. $$x^{2}+x y+y^{2}=3, \quad(1,1) \quad$$ (ellipse)

Use implicit differentiation to find an equation of the tangent line to the curve at the given point. \(x^{2}+2 x y-y^{2}+x=2, \quad(1,2)\) (hyperbola)

Use implicit differentiation to find an equation of the tangent line to the curve at the given point. \(x^{2}+y^{2}=\left(2 x^{2}+2 y^{2}-x\right)^{2}, \quad\left(0, \frac{1}{2}\right)\) (cardioid)

Use implicit differentiation to find an equation of the tangent line to the curve at the given point. \(x^{2 / 3}+y^{2 / 3}=4, \quad(-3 \sqrt{3}, 1)\) (astroid)

If \(V\) is the volume of a cube with edge length \(x\) and the cube expands as time passes, find \(d V / d t\) in terms of \(d x / d t\) .

(a) If \(A\) is the area of a circle with radius \(r\) and the circle expands as time passes, find \(d A / d t\) in terms of \(d r / d t\) . (b) Suppose oil spills from a ruptured tanker and spreads in a circular pattern. If the radius of the oil spill increases at a constant rate of 1 \(\mathrm{m} / \mathrm{s}\) , how fast is the area of the spill increasing when the radius is 30 \(\mathrm{m} ?\)