Diagnostic — checks Unit 2 (rules) + geometry/formula prereqs
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Purpose: Probes Unit 2 rule fluency, exponent algebra, and geometric formulas (cones, cylinders, spheres, Pythagoras) needed for related rates and optimization.
Q1 Prereq — Rules
[3]
Differentiate \(f(x)=x^3-6x^2+9x+1\) and find all values of \(x\) where \(f'(x)=0\).
Q2 Prereq — Geometry
[3]
State volume and surface-area formulas for: a) sphere, b) cone, c) cylinder. Identify which formula uses \(r^2\), \(r^3\), and \(rh\).
Q3 Prereq — Pythagoras
[2]
A 13 m ladder leans against a wall. The base is 5 m from the wall. How tall up the wall does it reach? Differentiate \(x^2+y^2=169\) implicitly with respect to time.
Q4 Thinking — Setup
[3]
Two numbers add to 12. Their product is to be maximized. Set up the function to optimize and state its domain. Do NOT solve.
Q5 Communication
[3]
Explain in your own words the difference between an ABSOLUTE maximum and a LOCAL maximum on a closed interval. Why must we check endpoints?
Q6 Thinking
[3]
For a particle with \(s(t)=t^3-3t^2\), interpret physically: a) when is \(v(t)>0\)? b) when is the particle accelerating in the positive direction? c) when is it speeding up?
Q7 Reflection
[2]
Which application is most challenging for you so far: motion, related rates, or optimization? What specifically?
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