Question

a) Which of the lines A, B or C shows the best estimate for the tangent to the curve at * = 5?
b) Use your answer to part a) to estimate the gradient of the curve at æ = 5.
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Answer 1

Answer:

  a) Line C

  b) -3

Step-by-step explanation:

You want to identify the tangent line to the given curve at x=5, and its approximate slope.

a) Tangent

A tangent line touches the curve at the point of tangency, and has the same slope as the curve at that point. It does not cross the curve there, but may cross the curve elsewhere, depending on the function's curvature.

The only line shown as touching the curve at one point is Line C. That point has x-coordinate x=5, so is the tangent we want.

b) Slope

It usually works best to estimate the slope of a graphed line if points on the line can be identified where it crosses grid intersections. We don't see such points for Line C. The x-value at the top of the graph (y=9) is about x=4.1, and at the bottom (y=0) is about x=7.1. These points let us compute the approximate slope as ...

  m = (y2 -y1)/(x2 -x1) = (0 -9)/(7.1 -4.1) = -9/3 = -3

The gradient of the curve at x=5 is about -3.