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Review questions for exam ii of math 105, covering topics such as finding derivatives, evaluating limits, and determining tangent lines. It includes exercises on various functions and curves.
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Math 105: Review for Exam II
(a) y = x^2 + 2x^ + e^2 + e^2 x^ + ln 2 + ln (2x) + arctan 2
(b) y =
x · arctan (5x)
(c) y = ln(tan(2cos(x
(^2) ) ))
(d) y = sin^3
x + eπ ln 4 + arcsin 6x
(e) y = (x^2 + 1)sin^ x
xy (known as the Folium of Descartes).
(a) Find dy/dx.
(b) Find the equation of the tangent line at the point (1,2).
(a) lim x→ 0
sin 3x 5 x
(b) lim x→∞
ex ln x
(c) lim x→ 0
1 − cos 2x 3 x
(d) lim x→ 1
x^3 − 1 7 − 7 x
(e) lim x→ 0
1 − cos 4x 5 x^2
(a) an antiderivative of y =
1 − 9 x^2
(b) tan(arccos x)
π
arcsin t+y^2 and that f
. Find the equation of the tangent line to f at
(a) If f′(1) = 0 then f always/sometimes/never has a critical point at x = 1.
(b) If f′(2) = 0 then f always/sometimes/never has a local maximum or local minimum at x = 2.
(c) If x = 3 is a critical point of f, then f′^ (3) is always/sometimes/never 0.
(d) If f′′(4) = 0, then f always/sometimes/never has an inflection point at x = 4.
(e) If f has a global maximum at x = 5, then f′^ (5) is always/sometimes/never 0.
(f) If f′(6) = 0 and f′′(6) = −2, then f always/sometimes/never has a local maximum at x = 6.
(g) If f′(7) = 0 and f′′(7) = 0, then f always/sometimes/never has a local extremum at x = 7.
