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Calculus 1 - Exam 1 Practice: Inverse, Exponential, Limits, Derivatives (90 characters) - , Exams of Calculus

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.

Typology: Exams

Pre 2010

Uploaded on 08/16/2009

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Math 111S โ€“ Calculus 1 Practice Problems for Exam 1 Fall 2008
1. Find the inverse of the function f(x) = 1+x
1โˆ’x.
2. Find the exponential function whose graph passes through the points (1,2) and (โˆ’1,9
2).
3. Given
f(x) = ln (xโˆ’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.
4. For each of the following functions, state the domain and range.
(a) f(x) = ln x
(b) g(x) = 1
(x+3)2
(c) The inverse of ยก1
3ยขx
5. For each of the following, solve for x.
(a) log2x= 4
(b) 3(x3โˆ’1) =b, where bis a positive constant.
(c) e3x=โˆ’1
(d) ln x+ ln(xโˆ’3) = 2 ln 2
6. A bacteria population doubles every 16 hours. Suppose that the initial population is
200.
(a) How many bacteria are present after 32 hours?
(b) Find a formula for p(t), the size of the population after thours.
(c) At what time tare there 5000 bacteria present? (You do not need to simplify
your answer!)
1
pf3

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Math 111S โ€“ Calculus 1 Practice Problems for Exam 1 Fall 2008

  1. Find the inverse of the function f (x) = 1+ 1 โˆ’xx.
  2. Find the exponential function whose graph passes through the points (1, 2) and (โˆ’ 1 , 92 ).
  3. Given

f (x) = ln (x โˆ’ 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.

  1. 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

  1. For each of the following, solve for x.

(a) log 2 x = 4 (b) 3(x (^3) โˆ’1) = b, where b is a positive constant. (c) e^3 x^ = โˆ’ 1 (d) ln x + ln(x โˆ’ 3) = 2 ln 2

  1. 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!)

  1. Let f be a function that satisfies all of the given conditions:

lim xโ†’โˆ’ 2 f (x) = 1, f (โˆ’2) = 3, lim xโ†’ 0 f (x) = 0, f (0) = 0

lim xโ†’ 1 โˆ’^

f (x) = 0, lim xโ†’ 1 +^

f (x) = โˆ’ 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.

  1. 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 = โˆž)

(a) lim xโ†’ 2 โˆ’

x โˆ’ 2

(b) lim xโ†’ 2

x^2 โˆ’ 4 x โˆ’ 2

(c) lim xโ†’ 0

|x| x

(d) lim xโ†’ 1 +

x^2 โˆ’ 1 |x โˆ’ 1 |

(e) lim xโ†’ 1 โˆ’

x^2 โˆ’ 1 |x โˆ’ 1 |

(f) lim xโ†’ 1

x^2 โˆ’ 1 |x โˆ’ 1 |

(g) lim xโ†’โˆ’โˆž

x^3 + 10x + 3 4 x^3 + x^2

(h) lim hโ†’ 0

(2 + h)^2 โˆ’ 4 h

(i) lim xโ†’โˆ’ 1

(x + 1)^4

(j) lim xโ†’ 1

x^2 โˆ’ 5 x^2 + 2

(k) lim xโ†’โˆž

3 x โˆ’ 1 (x + 1)(x โˆ’ 2)

(l) lim tโ†’ 1

t โˆ’ 1

t (t โˆ’ 1)

(m) lim xโ†’ 0 x^2 cos

x

(n) lim xโ†’ 1

x^2 + 1 x โˆ’ 1

(o) lim xโ†’ 1 f (x), wheref (x) =

x^2 + 1 x > 1 x โˆ’ 1 x โ‰ค 1

(p) lim xโ†’โˆž

16 x^4 + 2 x^2 โˆ’ 1

(q) lim tโ†’โˆž

sin t t

(r) lim xโ†’โˆ’โˆž

9 x^6 โˆ’ 7 x + 3 4 x^3 + 2

  1. Let f (x) = (^1) x and g(x) = cos x.

(a) Find a formula for the function f โ—ฆ g and g โ—ฆ f. (b) On what intervals is the function f โ—ฆ g continuous? g โ—ฆ f?

  1. (a) Consider the function f (x) = x (^2) + xโˆ’ 2.^ You notice that^ f^ (0) =^ โˆ’

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.