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Practice problems for exam 1 in math 111s - calculus 1, focusing on inverse functions, exponential functions, limits, and derivatives. Students are asked to find inverse functions, sketch graphs, determine domains and ranges, solve logarithmic equations, and find limits. Some problems involve finding the derivative of a function and the equation of the tangent line.
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Math 111S โ Calculus 1 Practice Problems for Exam 1 Fall 2008
(a) Sketch a graph of f. The sketch does not have to be perfect, but it should have the correct shape, and the line x = 2 should be clearly indicated.
(b) What are the domain and range of f?
(c) Find the inverse of f , or explain why it does not exist.
(a) f (x) = ln x (b) g(x) = (^) (x+3)^12
(c) The inverse of
3
)x
(a) log 2 x = 4 (b) 3(x (^3) โ1) = b, where b is a positive constant. (c) e^3 x^ = โ 1 (d) ln x + ln(x โ 3) = 2 ln 2
(a) How many bacteria are present after 32 hours? (b) Find a formula for p(t), the size of the population after t hours. (c) At what time t are there 5000 bacteria present? (You do not need to simplify your answer!)
lim xโโ 2 f (x) = 1, f (โ2) = 3, lim xโ 0 f (x) = 0, f (0) = 0
lim xโ 1 โ^
f (x) = 0, lim xโ 1 +^
f (x) = โ 1.
(a) At each of the values, -2, 0, and 1, decide if f is continuous or discontinuous and state your reasons in terms of the limits given above. (b) Sketch the graph of an example of a function satisfying the conditions given above.
(a) lim xโ 2 โ
x โ 2
(b) lim xโ 2
x^2 โ 4 x โ 2
(c) lim xโ 0
|x| x
(d) lim xโ 1 +
x^2 โ 1 |x โ 1 |
(e) lim xโ 1 โ
x^2 โ 1 |x โ 1 |
(f) lim xโ 1
x^2 โ 1 |x โ 1 |
(g) lim xโโโ
x^3 + 10x + 3 4 x^3 + x^2
(h) lim hโ 0
(2 + h)^2 โ 4 h
(i) lim xโโ 1
(x + 1)^4
(j) lim xโ 1
x^2 โ 5 x^2 + 2
(k) lim xโโ
3 x โ 1 (x + 1)(x โ 2)
(l) lim tโ 1
t โ 1
t (t โ 1)
(m) lim xโ 0 x^2 cos
x
(n) lim xโ 1
x^2 + 1 x โ 1
(o) lim xโ 1 f (x), wheref (x) =
x^2 + 1 x > 1 x โ 1 x โค 1
(p) lim xโโ
16 x^4 + 2 x^2 โ 1
(q) lim tโโ
sin t t
(r) lim xโโโ
9 x^6 โ 7 x + 3 4 x^3 + 2
(a) Find a formula for the function f โฆ g and g โฆ f. (b) On what intervals is the function f โฆ g continuous? g โฆ f?
1 2 <^ 0 and^ f^ (3) = 10 > 0 but f is never equal to 0. Explain why this does not contradict the Intermediate Value Theorem.