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MATH 105 Final Exam Review I April 10, 2004
1.) a) Using the limit definition of the derivative, compute f ′(x) if f (x) = (^5) −^3 2 x.
b) Find the equation of the tangent line to f at x = 2.
2.) Given that f (0) = 2, g(0) = 3, f ′(0) = 5, g′(0) = 7, and f ′(3) = π compute the following. a) h′(0) if h(x) = f (x)g(x)
b) j′(0) if j(x) = f g^ ((xx))
c) k′(0) if k(x) = f (g(x))
3.) a) Sketch a graph of a continuous function whose derivative is discontinuous at exactly two points.
b) Sketch a graph of a function which is always positive and decreasing and which satisfies the following: x→−∞lim f^ (x) =^ ∞;^ xlim→∞ f^ (x) = 2;^ xlim→ 1 −^ f^ (x) = 5;^ xlim→ 1 +^ f^ (x) = 4
4.) Compute dy/dx for each of the following. a) y = x^2004 + 2004x^ + e^2004 + 2004 x + ln (2004x) + arctan (2004x) + ln(2004)
b) y = √x cos(7x^3 )
c) y = e x (^) + π sin 4 − 7 x
d) y = tan (ex^2 arcsin(5x))
e) y^3 + yx^2 + x^2 = 3y^2 (trisectrix)
5.) Evaluate the following limits.
a) (^) xlim→∞^ x 2 ln x b) (^) xlim→ 0 sin (12xx 3 )^ −^12 x c) (^) xlim→ 0 e
x (^) − 1 cos x d) (^) xlim→ (^5352) x^ −−^710 x e) (^) xlim→ 0 −^1 x f) (^) xlim→ (^0) x^1
8.) For the graph of f shown, carefully sketch a graph of f ′^ on the axes below. As an aid, fill in the table below and make sure your graph agrees with the entries in the table. f positive negative increasing decreasing concave up concave down f ′
9.) The graph shown is of f ′, NOT f. At which labelled point is a) f greatest? b) f least? c) f ′^ greatest? d) f ′^ least? e) f ′′^ greatest? f) f ′′^ least? g) f increasing most rapidly? h) f decreasing most rapidly? On what interval(s) is i) f increasing? j) f ′^ increasing? k) f concave up?
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