Free response with a graphing calculator (Part A)
Elevation of a hiking trail
- Units 3, 4, 5, 6 and 8
- 9 points
- About 15 minutes
You can use a calculator on this question, just like on exam day.
A multi-part problem, usually set in a real-world context, where you need a graphing calculator for things like definite integrals, derivatives at a point and solving equations. You show your setup, give decimal answers to three places, and explain or justify your conclusions. On the exam: 2 questions in Part A of the free-response section (30 minutes, graphing calculator required). The whole free-response section is 6 questions in 90 minutes and counts for 50% of the score. AB and BC use the same format; on BC, one of these is usually a parametric, polar or vector question.
The question
For 0 ≤ x ≤ 6, the elevation of a hiking trail at a point x kilometers (measured horizontally) from the trailhead is E(x) meters, where E is a differentiable function with E(0) = 500 and E′(x) = 150 sin(x²/5)·e^(−x/6) meters per kilometer. (Your calculator should be in radian mode.)
Suggested time: 15 minutes
Your answers are saved in this browser as you type.
Part (a)
2 pointsFind all values of x, for 0 < x < 6, at which E has a relative minimum. Justify your answer.
0 / 2,500 characters
Part (b)
3 pointsFind the greatest elevation of the trail for 0 ≤ x ≤ 6. Justify your answer.
0 / 2,500 characters
Part (c)
2 pointsA hiker walks along the trail. Her horizontal distance from the trailhead at time t hours is x(t) = 6t/(t + 2) kilometers. Find the rate at which the hiker's elevation is changing, in meters per hour, at time t = 1.
0 / 2,500 characters
Part (d)
2 pointsAt what value of x, for 0 ≤ x ≤ 6, is the trail climbing most steeply (that is, where is E′(x) greatest)? Give a reason for your answer.
0 / 2,500 characters
Checking scoring…
Scoring it yourself shows you the rubric, examples and a model answer. Try writing your answer first.