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Estimate the Tangent Line and Gradient of a Curve at x=5
Mathematics
Grade 10 (Junior High)
Question Content
a) Which of the lines A, B or C shows the best estimate for the tangent to the curve at $x = 5$? b) Hence, estimate the gradient of the curve at $x = 5$
Correct Answer
a) Line A; b) $-1.25$ (approximate, range $-1.2$ to $-1.3$ is acceptable)
Detailed Solution Steps
1
Step 1: Identify the tangent line at $x=5$: A tangent line touches the curve at only one point and matches the curve's slope at that point. Line B is too shallow, Line C is too steep, so Line A is the correct tangent.
2
Step 2: Calculate the gradient of Line A: Pick two clear points on Line A, e.g., $(0,10)$ and $(8,0)$. Use the gradient formula $m = \\frac{y_2 - y_1}{x_2 - x_1}$.
3
Step 3: Substitute the points into the formula: $m = \\frac{0 - 10}{8 - 0} = \\frac{-10}{8} = -1.25$.
Knowledge Points Involved
1
Tangent to a Curve
A tangent line to a curve at a specific point is a straight line that touches the curve only at that point and has the same instantaneous slope as the curve at that location. It is used to approximate the curve's behavior near the point.
2
Gradient (Slope) of a Straight Line
The gradient measures the steepness of a line, calculated as $m = \\frac{\\text{change in } y}{\\text{change in } x} = \\frac{y_2 - y_1}{x_2 - x_1}$ for two points $(x_1,y_1)$ and $(x_2,y_2)$ on the line. A negative gradient means the line slopes downward from left to right.
3
Estimating a Curve's Slope Using Tangents
Since the slope of a curve changes at every point, the tangent line at a point gives the exact slope of the curve at that point. We estimate this by drawing the best-fit tangent and calculating its gradient.
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