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Math 111S โ Calculus 1 Practice Problems for Exam 1 Fall 2008
- Find the inverse of the function f (x) = 1+ 1 โxx.
- Find the exponential function whose graph passes through the points (1, 2) and (โ 1 , 92 ).
- Given
f (x) = ln (x โ 2) + 3.
(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.
- For each of the following functions, state the domain and range.
(a) f (x) = ln x (b) g(x) = (^) (x+3)^12
(c) The inverse of
3
)x
- For each of the following, solve for 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 bacteria population doubles every 16 hours. Suppose that the initial population is
(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!)
- Let f be a function that satisfies all of the given conditions:
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.
- For each of the following, either find the limit or infinite limit or explain why it does not exist: (If the limit is infinite then write = โ)
(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
- Let f (x) = (^1) x and g(x) = cos x.
(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?
- (a) Consider the function f (x) = x (^2) + xโ 2.^ You notice that^ f^ (0) =^ โ
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.
