Math  /  Calculus

Questionsurse ilctvo.arg/content/enforced/23228188-MCV4U-EN-02-03-ON-(I-D-0323)/course_content/assets/locker_docs/mcv4u_01.04.04.html?ou=23228188 ACA's Paralegal Dro.
3. Explain the difference between a secant line and a tangent line. How do they relate to the rate of change of a function? Include a sketch of each type of line in your solution. [ 6 marks]

Studdy Solution

STEP 1

1. A secant line intersects a curve at two or more points.
2. A tangent line touches a curve at exactly one point.
3. The rate of change of a function can be analyzed using both secant and tangent lines.
4. A sketch can visually represent the difference between a secant line and a tangent line.

STEP 2

1. Define the secant line.
2. Define the tangent line.
3. Explain the relationship between the secant line and the average rate of change.
4. Explain the relationship between the tangent line and the instantaneous rate of change.
5. Provide sketches to illustrate secant and tangent lines.

STEP 3

Define the secant line.
A secant line to a curve is a straight line that intersects the curve at two or more distinct points.

STEP 4

Define the tangent line.
A tangent line to a curve is a straight line that touches the curve at exactly one point and has the same slope as the curve at that point.

STEP 5

Explain the relationship between the secant line and the average rate of change.
The slope of the secant line between two points (x1,y1) (x_1, y_1) and (x2,y2) (x_2, y_2) on a function f(x) f(x) represents the average rate of change of the function over the interval [x1,x2][x_1, x_2]. Mathematically, it is given by:
Slope of secant line=f(x2)f(x1)x2x1 \text{Slope of secant line} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}

STEP 6

Explain the relationship between the tangent line and the instantaneous rate of change.
The slope of the tangent line at a point (x,f(x)) (x, f(x)) on a function f(x) f(x) represents the instantaneous rate of change of the function at that point. This is equivalent to the derivative f(x) f'(x) at that point.

STEP 7

Provide sketches to illustrate secant and tangent lines.
To illustrate, consider the function f(x) f(x) .
- The secant line between points A A and B B (e.g., A=(x1,f(x1)) A = (x_1, f(x_1)) and B=(x2,f(x2)) B = (x_2, f(x_2)) ) intersects the curve at these points. - The tangent line at point P P (e.g., P=(x,f(x)) P = (x, f(x)) ) touches the curve only at point P P .
\begin{array}{c} \text{Secant Line:} \\ \includegraphics[scale=0.5]{secant_line.png} \\ \text{Tangent Line:} \\ \includegraphics[scale=0.5]{tangent_line.png} \end{array}

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