Let f (x) = x5 + x2. The secant line between x = 0 and x = 1 has slope 2 (check this), so by the MVT,f'(c) = 2 for some c ∈ (0, 1). Plot f and the secant line on the same axes. Then plot y = 2x + b for different values ofb until the line becomes tangent to the graph of f . Zoom in on the point of tangency to estimate the x-coordinate c of thepoint of tangency.
Rate of Change
The relation between two quantities which displays how much greater one quantity is than another is called ratio.
Slope
The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:
Let f (x) = x5 + x2. The secant line between x = 0 and x = 1 has slope 2 (check this), so by the MVT,f'(c) = 2 for some c ∈ (0, 1). Plot f and the secant line on the same axes. Then plot y = 2x + b for different values of
b until the line becomes tangent to the graph of f . Zoom in on the point of tangency to estimate the x-coordinate c of the
point of tangency.
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