Problem 4
\(\mathrm{T} / \mathrm{F}:\) If \(\frac{d y}{d x}=0\) at \(t=t_{0},\) then the normal line to the curve at \(t=t_{0}\) is a vertical line.
Problem 4
Explain why the following is true: "If the coefficient of the \(x^{2}\) term in the equation of an ellipse in standard form is smaller than the coefficient of the \(y^{2}\) term, then the ellipse has a horizontal major axis."
Problem 5
Plot the points with the given polar coordinates. (a) \(A=P(2,0)\) (b) \(B=P(1, \pi)\) (c) \(C=P(-2, \pi / 2)\) (d) \(D=P(1, \pi / 4)\)
Problem 5
Find: (a) \(\frac{d y}{d x}\) (b) the equation of the tangent and normal lines to the curve at the indicated \(\theta\) -value. \(r=1+\sin \theta ; \quad \theta=\pi / 6\)
Problem 5
Explain how one can quickly look at the equation of a hyperbola in standard form and determine whether the transverse axis is horizontal or vertical.
Problem 5
Parametric equations for a curve are given. (a) Find \(\frac{d y}{d x}\). (b) Find the equations of the tangent and normal line(s) at the point(s) given. (c) Sketch the graph of the parametric functions along with the found tangent and normal lines. \(x=t, y=t^{2} ; \quad t=1\)
Problem 5
Sketch the graph of the given parametric equations by hand, making a table of points to plot. Be sure to indicate the orientation of the graph. \(x=t^{2}+t, \quad y=1-t^{2}, \quad-3 \leq t \leq 3\)
Problem 6
Plot the points with the given polar coordinates. (a) \(A=P(2,3 \pi)\) (b) \(B=P(1,-\pi)\) (c) \(C=P(1,2)\) (d) \(D=P(1 / 2,5 \pi / 6)\)
Problem 6
Find: (a) \(\frac{d y}{d x}\) (b) the equation of the tangent and normal lines to the curve at the indicated \(\theta\) -value. \(r=1-3 \cos \theta ; \quad \theta=3 \pi / 4\)
Problem 6
Parametric equations for a curve are given. (a) Find \(\frac{d y}{d x}\). (b) Find the equations of the tangent and normal line(s) at the point(s) given. (c) Sketch the graph of the parametric functions along with the found tangent and normal lines. \(x=\sqrt{t}, y=5 t+2 ; \quad t=4\)
