Harold Washington College Math 209 Test 2 Name Date 2/21/202
Harold Washington Collegemath 209test 2name Date 22120121 Show T
Show that the curvature of the helix r(t) =< a cos t, a sin t, bt > with a, b ≠ 0 is a / (a² + b²), and also that the unit tangent vector of this curve makes a constant angle with the z-axis whose value is θ = cos⁻¹ [b / √(a² + b²)].
Show that the representation of r(s) =< s + √(s² + 1), 1/2 (s + √(s² + 1)), √2 ln(s + √(s² + 1)) > is the arc length parametrization of the curve. In other words, show that || dr(s)/ds || = 1. Hint: Define u = s + √(s² + 1) and apply the chain rule.
Show that the parametrization of the curve r(t) =< e^t cos t, e^t sin t, e^t > with respect to arc length measured from t=0 in the direction of increasing t is r(s) = (s + √3) < sin(ln(s + √3)), cos(ln(s + √3)), 1 >.
Show that the curve r(t) =< t, 1 + t², 1 - t² > lies in a plane.
Show that the curvature along the curve r(t) =< t - sin t, 1 - cos t, t > is |κ| = √(1 + 4 sin⁴ t) / √(1 + 4 sin² t). Show that the torsion along the curve r(t) =< t - sin t, 1 - cos t, t > is τ = -1 / (1 + 4 sin⁴ t).
For the curve r(t) =< 3t - t³, 3t², 3t + t³ >, use the formulas |κ| = |r'(t) × r''(t)| / |r'(t)|³ and τ = |r'(t) × r''(t) · r'''(t)| / |r'(t) × r''(t)|² to show that |κ| = τ = t² / 4.
Show that the torsion of the helix r(t) =< a cos t, a sin t, bt > with a > 0 and b ≠ 0 is b / (a² + b²).
Show that the equation of the osculating plane for the curve r(t) =< t, t², t³ > is 3x - 3y + z = 1.
For the helix r(t) =< a cos t, a sin t, bt > with a, b ≠ 0 and t = π/3, find:
- The tangent vector, the normal vector, and the binormal vector.
- The tangent line, the normal line, and the binormal line.
- The rectifying plane, the normal plane, and the osculating plane.
Two particles travel along the space curves r₁(t) =< t, t², t³ > and r₂(t) =< 1 + 2t, 1 + 6t, 1 + 14t >. Do the particles collide? Where? Do their paths intersect? Where?
Find the intersection point of the helix r₁(t) =< cos t, sin t, t > and the curve r₂(t) =< 1 + t, t², t³ >, and find the angle of intersection at that point.
Evaluate the following limits:
- lim t→0 sin 10t sin 3t / (csc t - cot t)
- lim t→0 (e^t - e⁻t - 2t) / sin t
At what point do the curves r₁(t) =< t, 1 - t, 3 + t² > and r₂(s) =< 3 - s, s - 2, s² > intersect?
For the space curve r(t) =< t + t³/3, t - t³/3, t² >, determine:
- (a) the unit tangent vector,
- (b) the curvature,
- (c) the principal normal,
- (d) the binormal vector,
- (e) the torsion.
Find the length of the curve r(t) =< 3 cos t, 4 cos t, 5 cos t > from t=0 to t=2π.
Evaluate the following integrals:
- ∫ (sin(ln x), √(t² - 9 t³), t² - t + 6 t³ + 3 t) dt
- ∫ (1/√(1 + t²) ln(x + √(1 + t²)), t - √(tan⁻¹ t) / (1 + 4 t²), 1 / √ t(1 + t)) dt
If r(t) =< 3 x²(1 + x) - 9 x(1 + x) + 27(1 + x), (2 x² - 1) √(x² + 1) / 3 x³, 2 (3 a x - 2 b) √(a x + b) >, then r'(t) =< x²(1 + x) / 3, 1 / x⁴ √(x² + 1), x √(a x + b) >.
If r(t) =< x² / 4 + x sin 2x / 4 + cos 2x / 8, sin³ x / 3 - sin⁵ x / 5, x³ cos x - 3 x² sin x - 6 x cos x + 6 sin x >, then r'(t) =< x cos² x, sin² x cos³ x, -x³ sin x >.