A car travels along a straight road for 10 seconds

MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation

Problem

A car travels along a straight road for 10 seconds. Three smooth curves –

S

V

, and

W

– show measurements taken during the trip, all plotted on the same set of axes with time

t

(seconds) on the horizontal axis. Curve S •

S(0)=0

and

S

is increasing for

0\le t\le 10

. •

S

has a horizontal tangent at

t=0

and the magnitude of its slope increases until

t=5

, after which the slope decreases. • The graph of

S

is concave down on

(0,5)

and concave up on

(5,10)

. Curve V •

V(0)=0.8\,\text{m/s}

. •

V

decreases on

(0,5)

, reaching

V(5)=0

, then increases on

(5,10)

V(10)=0.8\,\text{m/s}

. Curve W •

W(t)<0

for

0<t<5

and

W(t)>0

for

5<t<10

. •

W

is increasing for all

t

(0,10)

. Answer the following. (1) Part 1: (a) Assign each curve –

S

V

, and

W

– to one of the following quantities of the car: position, velocity, or acceleration. Clearly state your assignment. (2) Part 2: (b) For the time interval

0<t<10

, determine all subintervals on which the car is speeding up. Justify your answer using relationships among the curves you identified in Part 1. (3) Part 3: (c) Explain why the car’s speed is minimal at

t=5

seconds, citing at least two features visible in the described graphs.

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