Free response·Challenge (5/5)·No calculatorNeeds mentor review

Difficulty guide

The label describes the problem, not your ability.

1/5 - Foundational

Warm-up setup; checks one basic idea.

2/5 - Routine exam skill

Normal syllabus skill; usually one main rule or interpretation.

3/5 - Standard multi-step

Exam-style problem with multiple steps or a common trap.

4/5 - Hard exam skill

High-end exam problem; combines methods carefully.

5/5 - Challenge

Stretch problem beyond normal exam pressure.

Question

A portion of the slope field for dydx=2xy\frac{dy}{dx}=\frac{2x}{y} is shown for y>0.y>0. Let y=f(x)y=f(x) be the solution with f(1)=3.f(1)=3.
Total: 6 points

(a) Find the slope of the solution at (1,3)(1,3) and write an equation for the tangent line there. Explain how the slope agrees with the field. (2 points)

(b) On the positive-yy portion of the slope field, where do solution curves have horizontal tangent segments? (1 point)

(c) Find the particular solution y=f(x)y=f(x). (2 points)

(d) Determine the largest interval of xx containing x=1x=1 on which this particular solution is differentiable. Justify your answer. (1 point)

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