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Solutions to exam i of math 111c, including calculations for logarithmic and limit problems, finding equations of asymptotes, and determining the slope of tangent lines. It covers topics such as logarithms, limits, functions, and calculus.
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September 28, 2004 Math 111 C — Exam I
Show all work clearly for partial credit. Do not use the graphing capabilities of your calculator.
(a) log 2 10 + 2 log 2 6 − log 2 45 (b)
ln 5
ln 5
(a) lim x→ 1
x^2 − 3 x + 2 x^2 − 1
(b) lim x→ 2
x^2 + 3x + 2 x^2 − 1
(c) lim x→ 0
ln(x^2 ) (d) lim x→ 0 −
( 1 |x|
x
)
(a) f (−2), (b) limx→− 2 f (x), (c) the equation(s) of the vertical asymptote(s), (d) the equation(s) of the horizontal asymptote(s), (e) limx→ 0 f (x), (f) the x-value(s) at which f has a removable discontinuity, and (g) the slope of the tangent line to the graph of y = f (x) at x = 1.
(a) f (x) =
x^2 − 4 x^2 − 3 x + 2
(b) g(x) = √^2 x x^2 + 4
2 x + 3 with the x-values x = 11 and x = 11 + h. (b) Using your answer to (a), find the slope of the tangent line to the graph of the function f (x) =
2 x + 3 at the point x = 11. (Use of a derivative formula — whatever that is — or simply the answer will receive no credit. Note: This is the same as the instantaneous rate of change of f (x) =
2 x + 3 with respect to x at the point where x = 11.) (c) Using your answer to (b), write the equation of the tangent line to the graph of√ f (x) = 2 x + 3 at x = 11.
x, a = 9 and ε = 0.4. What is the largest value of δ for which,
if 0 < |x − 9 | < δ , then |
x − 3 | < .4?
(An answer of the form “the smaller of the two numbers and ” is preferred but not required. The following drawing may be helpful.)
Some possibly useful equations:
y − y 0 = m(x − x 0 ) a^3 − b^3 = (a − b)(a^2 + ab + b^2 )
ar^ = b ⇐⇒ loga b = r loga c =
logb c logb a
(c) Because f (11) = 5, the desired tangent line is y − 5 = 15 (x − 11).
x is x = y^2 , the question marks are 2. 62 and 3. 42 ; so the answer is: the largest δ that works is the smaller of the two numbers 9 − 2. 62 [= 2.24] and 3. 42 − 9[= 2.56], [i.e., 2.24]. It would not have been necessary to give any of the material in the brackets.