Purpose: Probes MHF4U knowledge of exponent/log laws, special-angle trig values, and the chain rule.
Q1 Prereq — Log laws
[3]
Simplify using log laws (no calculator): a) \(\ln(e^5)\); b) \(\log_2 32\); c) \(\ln(e^{2x}\cdot e^{3x})\).
Q2 Prereq — Special angles
[3]
Without a calculator, state \(\sin,\cos,\tan\) for: \(0,\,\pi/6,\,\pi/4,\,\pi/3,\,\pi/2\).
Q3 Prereq — Identities
[2]
State the Pythagorean identity. Use it to write \(\cos^2 x\) in terms of \(\sin^2 x\). State the double-angle formula for \(\sin(2x)\).
Q4 Thinking
[3]
Predict (without computing): which is larger at \(x=2\), \(\frac{d}{dx}(2^x)\) or \(\frac{d}{dx}(e^x)\)? Why?
Q5 Communication
[3]
Explain in your own words why \(e\) is the "natural" base for exponentials. What special property makes \(\frac{d}{dx}e^x=e^x\)?
Q6 Thinking
[3]
Without computing the full derivative, state which derivative rules you'd use for: a) \(y=x^2 e^{x}\); b) \(y=\sin(3x^2)\); c) \(y=\frac{e^x}{\cos x}\). Just identify the rules.
Q7 Reflection
[2]
Which transcendental function (exp, trig, log) feels least familiar going into this unit? What concept is most unclear?
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