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This is the Exam of Calculus which includes Start Biking, Square Inch, Some Points, Solution, Slope, Slope Fields etc. Key important points are: Reals, Arctan, Curve Defined, Folium of Descartes, Point, Equation, Tangent Line, Limits, Point, Inverse Function
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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^5
x + eπ ln 4 + arcsin 6x
xy (known as the Folium of Descartes).
(a) Find dy/dx.
(b) Verify that the point (1,2) is on the curve above.
(c) Find the equation of the tangent line at the point (1,2).
(a) lim x→ 1
x^3 − 1 7 − 7 x
(b) lim x→ 0
1 − cos 2x 3 x
(c) lim x→ 0 +^
x^2 ln x
(d) lim x→ 0
1 − cos 4x 5 x^2
(e) lim x→∞
x^2 2 x
(a) What point must be on the graph of f−^1 (x)?
(b) What point must be on the graph of h(x)?
(c) Give an example of a point that cannot be on the graph of f(x). Do not choose a point with x-value of 2.
(d) What is the value of the derivative of h(x) at x = 2?
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.
(a) Write a differential equation whose solution is P (t).
(b) Solve your differential equation.
(c) When will the population reach 60000?
