Problem 8 - Calculus - AP Edition

Question

Sketch a curve and a line \(L\) satisfying the stated conditions. \(L\) intersects the curve in exactly one point, but \(L\) is not tangent to the curve.

Step-by-Step Solution

Answer

Draw a circle and a secant line just touching once without being tangent.

Step 1: Understanding the Problem

We need to find a curve and a line such that the line intersects the curve at exactly one point. Importantly, this intersection point must not be a point of tangency.

Step 2: Choosing the Curve

Select a simple curve such as a circle for ease of understanding. A circle is a closed curve where each direction leads back to the starting point, providing clarity in finding intersections.

Step 3: Choosing the Line

Consider a line that intersects the curve at only one point without being tangent. For a circle, a vertical or horizontal line (not going through the center) intersecting once on the circumference can fulfill the requirement.

Step 4: Verify Intersection Conditions

To confirm the line intersects the curve at exactly one point without tangency, we must ensure that at the intersection point, the line does not have the same slope as the tangent of the circle at that point.

Step 5: Sketching

Draw a circle and an appropriate line, ensuring that the line crosses through the circle at only one point. Verify visually that the line isn't tangent by checking that it doesn't just graze the circle's edge.

Key Concepts

Curve Intersection

When we talk about curve intersection, we refer to finding points where a line meets a curve. In our scenario, we are focusing on lines intersecting a curve at precisely one point. However, it is crucial that this intersection is not a point of tangency. The distinction is that in a typical intersection, the line and the curve meet but are not parallel.

• A classic method to visualize this is using a circle, a simple and balanced shape.
• Consider a line that grazes the circle at a single spot but doesn't run along it.

This line will intersect the circle once, which means they share just one point in common. Practically, this can be a line stopping at the circle's boundary without touching any tangents of the curve.

Tangency

Tangency occurs when a line touches a curve at exactly one point and does not cross the curve at that point. In technical terms, they have the same slope at the intersection point. The line appears to just barely touch the curve, and then continue without crossing it.

• For example, consider a tangent to a circle; it's a straight line that meets the circle at one exclusive spot with the line's direction matching that of the circle at that particular point.
• A curve and line in tangency often look like a point of intersection but with an added condition of matching slopes.

An important point is that in our exercise, the intersection isn't a tangency. That implies ensuring the line's slope at the intersection differs from the curve's slope there.

Sketching Techniques

Sketching curves and lines requires understanding the overall shapes and their mathematical properties. For curves like circles, knowing their symmetry helps immensely in drawing accurate sketches.

• Begin by drawing the circle, carefully marking its center and maintaining an equal radius all around to guide your pen or pencil in keeping the circle's integrity.
• Next, add the line. For our exercise, ensure it's positioned to hit only one side of the circle without being tangent.

Take into account aspects like the line's inclination and ensure it doesn’t align tangentially. Use light pencil strokes for preliminary sketches as adjustments might be necessary before finalizing. Paying attention to where exactly the line meets the curve will prevent mistakes, ensuring a true intersection.