Question 1

What is the equation of a tangent line?

Accepted Answer

The equation of a tangent line is a linear equation that represents a line touching a curve at exactly one point (locally). At the point of tangency, the tangent line has the same slope as the curve. The equation is typically written in slope-intercept form (y = mx + b) or point-slope form (y - y₀ = m(x - x₀)), where m is the derivative at the point and (x₀, y₀) is the point of tangency.

Question 2

How do you find the equation of a tangent line?

Accepted Answer

To find the equation of a tangent line: 1) Find the derivative f'(x) of the function. 2) Evaluate the derivative at x = x₀ to get the slope m = f'(x₀). 3) Find the y-coordinate: y₀ = f(x₀). 4) Use point-slope form: y - y₀ = m(x - x₀). 5) Simplify to slope-intercept form: y = mx + b. Example: For f(x) = x² at x = 2: f'(x) = 2x, m = 4, point (2, 4), equation: y = 4x - 4.

Question 3

What is the slope of a tangent line?

Accepted Answer

The slope of a tangent line at a point is equal to the derivative of the function evaluated at that point. It represents the instantaneous rate of change of the function. For a function f(x), the slope at x = a is m = f'(a). For example, if f(x) = x³, then f'(x) = 3x², so the slope at x = 2 is f'(2) = 3(2)² = 12. The slope tells you how steep the tangent line is and whether it's increasing or decreasing.

Question 4

What is point-slope form for tangent lines?

Accepted Answer

Point-slope form is y - y₀ = m(x - x₀), where m is the slope and (x₀, y₀) is a point on the line. For tangent lines: m = f'(x₀) (the derivative at the point) and (x₀, y₀) is the point of tangency. This form is ideal for tangent lines because you directly substitute the known point and slope. Example: For slope 3 at point (1, 2): y - 2 = 3(x - 1), which simplifies to y = 3x - 1.

Question 5

How to find tangent line for different function types?

Accepted Answer

For different functions: Polynomial (f(x) = x^n): Derivative is f'(x) = nx^(n-1). Trigonometric (f(x) = sin(x)): f'(x) = cos(x). Exponential (f(x) = e^x): f'(x) = e^x. Logarithmic (f(x) = ln(x)): f'(x) = 1/x. Product/Quotient: Use product rule or quotient rule. Chain rule: For composite functions. Always evaluate the derivative at the specific x-value to get the slope.

Question 6

What is the difference between tangent line and secant line?

Accepted Answer

Tangent line touches the curve at one point with the same slope as the curve at that point. It represents the instantaneous rate of change. Secant line connects two points on the curve and represents the average rate of change between those points. As the two points get closer, the secant line approaches the tangent line. This concept is fundamental to the definition of the derivative as a limit.

Question 7

How to use tangent line for linear approximation?

Accepted Answer

The tangent line provides a linear approximation of the function near the point of tangency. Formula: L(x) = f(x₀) + f'(x₀)(x - x₀). This approximates f(x) for x close to x₀. Example: For f(x) = √x at x₀ = 4: f(4) = 2, f'(x) = 1/(2√x), f'(4) = 1/4. Approximation: L(x) = 2 + (1/4)(x - 4). So √4.1 ≈ 2 + (1/4)(0.1) = 2.025. This is useful for estimating function values.

Question 8

What does it mean when a tangent line is horizontal?

Accepted Answer

A horizontal tangent line occurs when the derivative (slope) equals zero at that point: f'(x₀) = 0. This indicates a critical point—either a local maximum, local minimum, or saddle point. The equation is y = y₀ (a horizontal line at height y₀). Example: f(x) = x² has a horizontal tangent at x = 0 where f'(0) = 0, giving the line y = 0. Horizontal tangents are crucial for optimization problems.

Question 9

Can a function have multiple tangent lines at one point?

Accepted Answer

A differentiable function has exactly one tangent line at each point—the derivative gives a unique slope. However, non-differentiable functions can have: No tangent line (at corners or cusps like |x| at x = 0). Multiple tangent lines (at points where the function isn't well-defined). Vertical tangent lines (when derivative is undefined but limit approaches ±∞). For smooth, differentiable functions, there's always exactly one tangent line at each point.

Question 10

How is the equation of tangent line used in calculus?

Accepted Answer

Equation of tangent line is used in calculus for: 1) Linear approximation (estimating function values). 2) Newton's method (finding roots iteratively). 3) Optimization (identifying maxima and minima via horizontal tangents). 4) Related rates problems. 5) Differential equations (Euler's method). 6) Error analysis and bounds. 7) Curve sketching and behavior analysis. It's one of the most practical applications of derivatives in real-world problem solving.

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