Stewart – Calculus – 2.5 – Continuity

  1. Definition: A function f is continuous at a number a if

     

    This definition implicitly requires three things if f is continuous at a:

    1. f(a) is defined (that is, a is in the domain of f)

    2. \lim_{x \to a}f(x) exists

    3. \lim_{x \to a}f(x)=f(a)


  2. If f and g are continuous functions with f(3)=5 and
    \lim_{x \to 3}[2f(x) - g(x)]=4, find g(3).


    Use the definition of continuity and the properties of limits to show that the funtion is continous at the given number a. f(x)=(x+2x^{3})^{4}, a=-1


  3. The Intermediate Value Theorem
    Suppose that f is continuous on the closed interval [a, b] and let N be any number between f(a) and f(b), where f(a) \neq f(b). Then there exists a number c in (a,b) such that f(c)=N.

    ~ ~ ~

    Use the Intermediate Value Theorem to show that there is
    a root of the given equation in the specified interval.

    x^4+x-3=0, (1,2)

Calculus Test 1

  1. Specify the domain of the function y = |x – 1|.

  2. Which of the following is a true statement about the graph of the equation y = x^4 +1?
  3. 1. It is symmetric about the x-axis.
    2. It is symmetric about the y-axis.
    3. It has two x-intercepts.
    4. It has no x-intercepts.

    a. Only statements 2 and 3 are true.
    b. Only statements 2 and 4 are true.
    c. Only statements 1 and 3 are true.
    d. Only statement 2 is true.


  4. Which of the following statements are true of the graph of y = \frac{2x-1}{x+1} ?

    1. It has no x-intercept.
    2. It has a slant asymptote y=2x.
    3. It has a vertical asymptote at x=-1.
    4. It has a horizontal asymptote at y=2.

    a. Statements 1, 2, and 3 are true.
    b. Statements 3 and 4 are true.
    c. Statements 2, 3, and 4 are true.
    d. All four statements are true.


  5. Let f(x) = 2 \cos x. The domain of f^{-1}(x) is

    1. [-1,1]
    2. (2,  \infty)
    3. (-\infty,  \infty)
    4. (-2,2)
    5. [-2,2]


  6. Simplify as far as possible. \ln e +a^{\log_{a}5}- \log100+10^0 - \log_{3} \frac{1}{3}
  7. Solve for x.

    a. \log_3 x + \log_3(2x+5)=1

    b. \frac{1}{\sqrt{2}}=4^{x}


  8. f(x)=(x-3)(x+1)^2(x-1)^4
  9. Identify the parts of the following composite function. f(g(h(j(x))))=\frac{1}{\sqrt{\log(x-1)}}
  10. Consider the picture below. Find the equation of the line L.


  11. The following table represents a function of the form f(x)=ab^{x}. Find the equation of the function.

      \begin{tabular}{| l | c | r |}  \hline  x & f(x)\\ \hline    0 & 6\\ \hline    1 & 18\\ \hline    2 & 54\\ \hline    3 & 162\\ \hline    4 & 486\\  \hline  \end{tabular}


  12. Given a^{m}=2, a^{n}=3, b^{m}=4, and b^{n}=5,
    use the properties of exponentials to determine (a^{3n}b^{m+n})^{\frac{1}{3}}.


  13. Evaluate:
    1. \sin \frac{\pi}{3}

    2. \tan \frac{3\pi}{4}

    3. \cos \frac{5\pi}{6}


  14. Find all solutions to the equaion, \tan^{2}x=\tan x, such that x \in [0,2\pi].
    Express the answers in radians.


  15. Find f^{-1}(x) if f(x) =  \sqrt{e^{x}+2}.
all real numbers
b
b
5
6

1. Set x equal to 0 to find the y-intercept.
2. Set the values in parentheses equal to 0 to get the x-intercepts 3, —1, and 1.
4. Note that the function will not cross the x axis with an even power, but will cross with an odd power.

Stewart – Calculus – 1.6 – Inverse Functions and Logarithms

  1. Can two different numbers have the same cube?


  2. Find the inverse of the equation


  3. Find the inverse of the equation


  4. True or false:

    is equal to
    .


  5. Solve.


  6. Find the inverse function of


  7. What is the inverse of the equation log _{a}x=y?


  8. Find equivalent equations.


  9. Find the equations that correspond to the labels.


  10. Find equivalent forms.


  11. Use the laws of logarithms to evaluate log_2{80} - log_2{5}.


  12. Find the equivalent natural log.
    log_e{x}


  13. Find the equivalent equation for ln x = y.


  14. Find the equivalent forms.


  15. Solve the equation
    .


  16. Express  \ln a+\frac{1}{2}\ln b as a single logarithm.


  17. The change of base formula says that
    For any positive number  a(a{\neq}1), we have  log_a{x}=.


  18. Sketch the graph of the function y=\ln (x-2)-1.


  19. Find the inverse of  \sin ^{-1}x=y.


  20. Evaluate  \tan (\arcsin \frac{1}{3}).

No.
False.
  1. Re-write the equation:
  2. Solve this equation for x:
  3. Interchange x and y:
  4. So, inverse function is:





ln x






Let , so
Then we can draw a right triangle with angle θ as in the figure

and deduce from the Pythagorean Theorem that the third side has length

This enables us to read from the triangle that