A cyclist travels along a straight road

MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties

Problem

A cyclist travels along a straight road. A computer displays a smooth position-versus-time graph for the first 6 s of the ride. At

t = 4.0\,\text{s}

, a tangent line drawn to the curve intersects the time axis at

(6.0\,\text{s},\;0\,\text{m})

and the position axis at

(0\,\text{s},\;\!\!−12.0\,\text{m})

. A second tangent line is drawn at

t = 4.5\,\text{s}

; it intersects the time axis at

(6.5\,\text{s},\;0\,\text{m})

and the position axis at

(0\,\text{s},\;\!\!−13.5\,\text{m})

. Use this information to investigate the cyclist’s motion. (a) Using the first tangent line, calculate the cyclist’s velocity at

t = 4.0\,\text{s}

. Show your reasoning and include units. (b) Using both tangent lines, compute the average acceleration of the cyclist between

t = 4.0\,\text{s}

and

t = 4.5\,\text{s}

. Clearly show how you use the results from part (a) and the slope of the second tangent line. (c) Discuss how the spacing of the two points chosen on each tangent line (far from the point of tangency versus close to it) would influence the numerical value obtained for the slope, and hence the velocity. Include qualitative reasoning about graphical uncertainty.

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