Type 1: Finding the Tangent Line Given a Known Tangency Point
1. In case of an extreme value point of a function, the equation of the tangent line at that point is ______.
【Answer】
【Explanation】 Based on the given extreme value point, the derivative value (k) is determined. Then, using the geometric meaning of derivatives, the equation of the tangent line can be derived.
For instance, if the function's extreme value point implies k=e, then...
The tangency point is (1, -e), and hence...
The equation of the tangent line is... Therefore, the answer is:
Key Point: This problem primarily tests the understanding and application of the geometric meaning of derivatives and extremes.
Type 2: Finding the Tangent Given Its Slope
2. If the image of y=ax intersects with the line y=x at a single point, find the value of a.
Type 3: Tangent Lines Passing Through a Given Point
3. The curve y = 3 - 2x^2 can be tangent to a line passing through the point (1, 0). What is the value of ?
Answer: 1, 9, 0.
Setting the tangency point's x-coordinate as 0 in the curve equation and writing the tangent line equation, we get... When substituted with (1, 0), we have two solutions. There are two scenarios: one where the equation has a repeated root that is not zero, and another where the equation has two distinct roots, one of them being zero.
For a given function f(x) = -3x^3 + 2x' + 5 at x=0 (where prime denotes derivative), find the equation of the tangent line at (0, f(0)).
Solution: The derivative of f(x) is f'(x) = -6x^2 + 2. At x=0, f'(0) = 2. Therefore, the slope of the tangent line is 2. Using this slope and the point (0, f(0) = 5), we get the equation of the tangent line as y = 2x + 5.
4. For f(x) = x^3 - 3x,find the equations of the tangent lines passing through point P(2, 2).
Answer: = 9x - 16 or y = 2.
Explanation: When P is the tangency point, the slope of the tangent line is derived from f'(x) = 3x^2 - 3. At x=2, this gives a slope of 9. Using this slope and P(2, 2), we get one equation. For a different tangency point (m, m^3 - 3m), we set up a system based on f'(m) and m to derive another equation for the tangent line.
General Concepts in Tangent Line Problems:
The utilization of derivatives in finding tangent lines is based on two primary ideas: finding slopes and then utilizing these slopes to form equations for lines passing through given points or known points on curves.
5. If a straight line is tangent to a curve, then...
【Answer】
【Explanation】
This involves finding the slope of a line at a given point on a curve using derivatives. It entails finding the derivative at a particular point, using it to determine the slope of a tangent line at that point, and then formulating an equation for that line.
6. Two tangents are drawn from a point to a given curve. If these tangents intersect with the x-axis at two points, what would be...?