Answer:
6. A smooth curve is a curve in mathematics that has no abrupt corners, cusps, or discontinuities in its direction or curvature. More formally, a smooth curve is one that is continuously differentiable, meaning that its derivative exists and is continuous over its entire domain.
Examples of smooth plane curves:
- A circle: The equation of a circle in Cartesian coordinates is (x - a)^2 + (y - b)^2 = r^2, where (a, b) is the center of the circle and r is its radius. Circles have a constant curvature and direction, making them smooth curves.
- A parabola: The equation of a parabola in standard form is y = ax^2 + bx + c, where a, b, and c are constants. Parabolas have a smooth and continuous curvature, and they do not have abrupt corners.
- An ellipse: The equation of an ellipse in Cartesian coordinates is (x/a)^2 + (y/b)^2 = 1, where a and b are positive constants. Ellipses have a smooth and continuous curvature and do not have abrupt changes in direction.
Examples of plane curves that are not smooth:
- A corner of a square: If you consider just one of the corners of a square, it has a sharp, 90-degree angle. Sharp corners like this are not smooth because the direction and curvature change abruptly at the corner.
- A cusp in a curve: A cusp is a point on a curve where it changes direction abruptly, often forming a sharp point. An example is the cusp in the curve y = x^(2/3) at the origin (0,0).
7. Finding the arc length of a curve by integration involves breaking down the curve into infinitesimally small segments and summing up the lengths of these segments. Here are the steps to work out an example of finding the arc length of a curve using integration:
Step 1: Define the curve and its parametric equations.
Let's consider a parametrically defined curve in 2D space: x = f(t) and y = g(t), where a ≤ t ≤ b represents the parameter range.
Step 2: Find the derivative of the parametric equations.
Calculate dx/dt and dy/dt.
Step 3: Calculate the differential arc length element ds.
Use the Pythagorean theorem: ds = √[(dx/dt)^2 + (dy/dt)^2] dt.
Step 4: Set up the integral for the total arc length L.
Integrate ds over the parameter range [a, b]:
L = ∫[a, b] √[(dx/dt)^2 + (dy/dt)^2] dt.
Step 5: Evaluate the integral.
Use appropriate techniques, such as substitution, to evaluate the integral.
Step 6: Simplify and express the result in terms of t or any other variable if necessary.
Step 7: If the integral is complex, you can use numerical methods or software to approximate the value of the arc length.
This process allows you to find the length of a curve between two specified points on the curve. It is a fundamental concept in calculus and has applications in various fields, including physics, engineering, and geometry.
Explanation: