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Math 1A Review Problems: Functions, Domains, Ranges, Transformations, Inverses, Limits, Exams of Calculus

University of California - BerkeleyCalculus

Review problems for math 1a students, covering topics such as determining if a curve is the graph of a function, finding domains and ranges, transformations of functions, and finding inverse functions. It also includes problems on limits and continuity.

Typology: Exams

Pre 2010

Uploaded on 10/01/2009

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Math 1A Review Problems (1.1 - 2.6)
Marcos Charalambides
September 20, 2009
Disclaimer : Try as many of the following problems as you can/want. Do
them without looking at the book or your notes. Come see me during my
office hours if you would like to talk about any of the problems. Please note
that some problems are harder than others (and there may be a couple of hard
ones). I will post solutions to several of these problems online by Sunday.
1. Given a curve, explain how to tell whether it is the graph of a function.
Give (graphical) examples of functions and non-functions
2. Find the domain of:
(a) x
2x−5
(b) 25√t
(c) √t+√t−1
(d) √t+3
√t−1
(e) 1
4
√x2−7x
(f) 1
√2x2+2−√x2−3x
3. Find the range of:
(a) √9−x2
(b) 1
1+x2
(c) 6e−x2
(d) 4 sin(x2)
(e) cos(sin x)
4. Suppose the graph of fis given. Write equations for the graphs obtained
from fas follows:
(a) Shift 2 units downward.
(b) Reflect about the x-axis.
(c) Stretch horizontally by a factor of 5.
1
pf3

Partial preview of the text

Download Math 1A Review Problems: Functions, Domains, Ranges, Transformations, Inverses, Limits and more Exams Calculus in PDF only on Docsity!

Math 1A Review Problems (1.1 - 2.6)

Marcos Charalambides

September 20, 2009

Disclaimer : Try as many of the following problems as you can/want. Do them without looking at the book or your notes. Come see me during my office hours if you would like to talk about any of the problems. Please note that some problems are harder than others (and there may be a couple of hard ones). I will post solutions to several of these problems online by Sunday.

  1. Given a curve, explain how to tell whether it is the graph of a function. Give (graphical) examples of functions and non-functions
  2. Find the domain of: (a) (^2) xx− 5 (b) 25√t (c) √t + √t − 1 (d) √t + √^3 t − 1 (e) √ (^4) x (^21) − 7 x (f) √ 2 x (^2) +2−^1 √x (^2) − 3 x
  3. Find the range of: (a) √ 9 − x^2 (b) (^) 1+^1 x 2 (c) 6e−x^2 (d) 4 sin(x^2 ) (e) cos(sin x)
  4. Suppose the graph of f is given. Write equations for the graphs obtained from f as follows: (a) Shift 2 units downward. (b) Reflect about the x-axis. (c) Stretch horizontally by a factor of 5.
  1. Explain how to obtain the following graphs from that of y = f (x): (a) y = f (x − 25) (b) y = − 3 f (x) (c) y = 4f (x) − 1 (d) y = |f (x)|
  2. Start by sketching y = tan x. By applying transformations as in Section 1.3 sketch y = 18 tan(x + π 4 ).
  3. Find f + g, f g, f ◦ g and f ◦ f for the following pairs of functions: (a) f (x) = x + 1, g(x) = x − 1 (b) f (x) = x^2 − x, g(x) = ex−^1 (c) f (x) = ex^2 , g(x) = e−x^2 (d) f (x) = e^2 x, g(x) = ln(x 12 )
  4. Suppose that f and g are both even. Must f + g be even? How about f g? What can you say if both f and g are odd?
  5. (a) Suppose that g is even. Show that f ◦ g is even for any function f. (b) Suppose g is odd. Is f ◦ g always odd? What if f is also odd? (c) Suppose f is even and g is odd. What can you say about f ◦ g?
  6. Suppose you are given the graph of a one-to-one function f. Explain how to sketch the graph of the inverse function f −^1.
  7. Find the inverse of: (a) f (x) = x^6 + 1, x ≥ 0 (b) f (x) = 2 − e^3 x
  8. Explain why f (x) = 32 + 2x^3 + 12 ex−^1 is one-to-one. Find f −^1 (4).
  9. Suppose that a 6 = b and A, B are non-zero constants. Solve Aeax^ = Bebx^ for x.
  10. Simplify cos(2 tan−^1 x).
  11. Express ln(1 + x^4 ) − 13 ln(x) + ln(cos x) as a single logarithm.
  12. Explain what is meant by a ”vertical asymptote”. If limx→a− f (x) = −∞ then can we necessarily conclude that f has a vertical asymptote? What if limx→∞ f (x) = −∞?
  13. Write down the precise definition for: (a) limx→a f (x) = L