YOU CAN PASS~!!!! MATH EXAM 1
decreasing
A function f is __________ on an interval I if:
increasing
A function f is __________ on an interval I if:
x^a
Complete the fact about power functions: Functions of the form f(x)=______, where a is constant.
1
Consider the function below. Use it to evaluate each of the following expressions. (If an expression does not exist, enter NONE.) a) lim g(x) x---->1^-
1
Consider the function below. Use it to evaluate each of the following expressions. (If an expression does not exist, enter NONE.) b) lim g(x) x----->1^+
5
Consider the function below. Use it to evaluate each of the following expressions. (If an expression does not exist, enter NONE.) c) lim g(1)
no
Does this function pass the horizontal line test?
yes
Does this function pass the horizontal line test?
no
Does this function pass the vertical line test?
yes
Does this function pass the vertical line test?
x=-3 x=4
Estimate the solutions of the equation f(x) = -1. (hint), you are now looking for x values. x= (smaller value ) x= (larger value)
0/20
Evaluate the limit, if it exists. (If it does not exist, enter NONE).
1/-49
Evaluate the limit, if it exists. (If it does not exist, enter NONE).
1/2
Evaluate the limit, if it exists. (If it does not exist, enter NONE).
18
Evaluate the limit, if it exists. (If it does not exist, enter NONE).
-49
Evaluate the polynomial. 2x^3 - x^2 - 4x + 2 at x = -3
polynomial rational function a domain a a
Fill in the Direct Substitution Property: If f(x) is a _____________ or a _______________ function AND ____ is in the ________ of f(x) then lim f(x) = f(?) x--->?
(a,b) b a N f(a) DNE f(b) c a b f(c) = N
Fill in the Intermediate Value Theorem. Suppose f is continous at ______ and left continuous at ___, right continuous at ____, and there is a number ___ between f(a) and f(b) where ___________ then there exists a number ____ between ___ and ____ such that _____________
x a close
Fill in the definition of a limit: The limit of f(x) as ____ approaches _____ is some number L if f(x) gets arbitrarily ______ to L as x gets arbitrarily close to a notation.
a^n b^n
Fill out the following exponent rule. (ab)^n=?
1/a^n
Fill out the following exponent rule. a^-n=?
1
Fill out the following exponent rule. a^0 = ?
a
Fill out the following exponent rule. a^1=?
a^mn
Fill out the following exponent rule. a^m*a^n =?
a^m-n
Fill out the following exponent rule. a^m/a^n=?
a^m*n
Fill out the following exponent rule. (a^m)^n =?
log(e)x
Fill out the following log rule. ln(x)=?
log base b (x)/log base b (a)
Fill out the following log rule. log (base) a (x) =?
x (same bases cancel)
Fill out the following log rule. log base b (b^x)=?
2x^2-32x+128
Find expressions for the quadratic function whose graph is shown.
x cannot equal -5 and 3
Find the Domain: (x+2) _____ (x^2+2x-15)
all values where x is not equal to 0
Find the Domain: 3/x
200ft/sec
Find the average velocity over [15,16] t s(t) ---------------------- 0 : 200 10 :500 15 :1000 16 : 1200 20 : 2100 30 : 3200 HINT: s(16)-s(15)/16-15
[5, infinity)
Find the domain of the function
(-infinity, -6)U(-6,4)U(4,infinity)
Find the domain of the function.
2-3(cos(x))
Find the following function. f(x) = 2 - 3x g(x) = cos(x) f o g =
cos(2-3x)
Find the following function. f(x) = 2 - 3x g(x) = cos(x) g o f =
2-3(2-3x)
Find the following functions. f(x) = 2 - 3x g(x) = cos(x) f o f =
cos(cos(x))
Find the following functions. f(x) = 2 - 3x g(x) = cos(x) g o g =
even
Find the symmetry of this function. f(x) = -2x^2+5
-1
For the function g whose graph is given, state the value of each quantity, if it exists. (If it does not exist, enter NONE.) a) lim g(t) x---->0^-
-2
For the function g whose graph is given, state the value of each quantity, if it exists. (If it does not exist, enter NONE.) b)lim g(t) x------>0^+
none (because there are two seperate values at x=0)
For the function g whose graph is given, state the value of each quantity, if it exists. (If it does not exist, enter NONE.) c)lim g(t) x----->0
3
For the function g whose graph is given, state the value of each quantity, if it exists. (If it does not exist, enter NONE.) d) lim g(t) x----->4
2
For the function g whose graph is given, state the value of each quantity, if it exists. (If it does not exist, enter NONE.) e) lim g(t) t---->2^-
0
For the function g whose graph is given, state the value of each quantity, if it exists. (If it does not exist, enter NONE.) f) lim g(t) t----->2^+
none
For the function g whose graph is given, state the value of each quantity, if it exists. (If it does not exist, enter NONE.) g) lim g(t) t-----> 2
1
For the function g whose graph is given, state the value of each quantity, if it exists. (If it does not exist, enter NONE.) h) lim g(2)
-2 and 2
For what values of x is f(x) = g(x)? Include the two values.
plug in numbers close to 3, find a reasonable guess
How could you numerically test this limit? x^2-9 -------- x-3 lim x---->3
finding its inverse
How do you find the symmetry of a function?
by the squeeze theorem, lim x^2sin1/x = 0 x--->0
How would you answer this Squeeze Theorem problem? lim x--->0 x^2sin(1/x)
odd
If f(-x) = -f(x), what is the symmetry of the function?
even
If f(x) = f(-x), what is the symmetry of the function?
odd
Non-symmetrical functions are:
x=3
Solve for x. 5^(3-2x)=5^-x
x=7e^2+1/3
Solve for x. ln(-3x-1)-ln7 = 2
4
State the value of the function given below: g(3)=
-2
State the values of function given below. f(-4)=
even
Symmetrical functions are:
2
The graphs of f and g are given. Use them to evaluate each limit, if it exists. If the limit does not exist, enter NONE. lim [f(x)+g(x)] x---->2
none
The graphs of f and g are given. Use them to evaluate each limit, if it exists. If the limit does not exist, enter NONE. lim [f(x)+g(x)] x-----> 1 HINT: Something plus DNE is DNE.
.333333
The point P(1, 1/2) lies on the curve y = x/(1 + x). If Q is the point (x, x/(1 + x)), use a scientific calculator to find the slope of the secant line PQ (correct to six decimal places) for the following value of x. a)0.5
.263158
The point P(1, 1/2) lies on the curve y = x/(1 + x). If Q is the point (x, x/(1 + x)), use a scientific calculator to find the slope of the secant line PQ (correct to six decimal places) for the following value of x. b)0.9
accurate it is
The smaller the interval of instantaneous velocity is, the more____________
2
This graph has been VERTICALLY stretched by how many units?
2 (because, 1.5, the originally value, has increased to 3, meaning that 1.5 * 2 will equal 3.)
This graph has been VERTICALLY stretched by how many units?
7
This graph has been shifted HORIZONTALLY by how many units?
6
Use the graph of the function f to state the value of each limit, if it exists. a) lim f(x) x----->0^-
0
Use the graph of the function f to state the value of each limit, if it exists. b) lim f(x) x----->0^+
DNE
Use the graph of the function f to state the value of each limit, if it exists. c) lim f(x) x----->0
2
Use the table to evaluate the expression. f(g(1))
y = 2sqrt(3(x-7)-(x-7)^2)
Using the original function y = sqrt(3x − x2), we can rewrite y = 2f(x − 7) as:
jump, removable, infinite
What are the three kinds of discontinuities?
1. F(x) is defined at f(a) 2.lim f(x) exists x-->a 3. they must be equal lim f(x) = f(a) x--->a
What are the three requirements for a function to be continuous?
ax^2+bx+c=y
What does a quadratic function look like?
|x|
What does an absolute value function look like?
change between the two points
What does the slope of a secant line represent?
reflect x axis
What does y = -f(x) do to a parent function?
increase the graph vertically by c units
What does y = cf(x) do to a parent function?
reflect y axis
What does y = f(-x) do to a parent function?
increase the graph horizontally by c units
What does y = f(cx) do to a parent function?
shifts up c units
What does y = f(x)+c do to a parent function?
shift down c units
What does y = f(x)-c do to a parent function?
shift left c units
What does y = f(x+c) do to a parent function?
shift right c units
What does y = f(x-c) do to a parent function?
difference quotient
What equation do we use to find the secant line?
a line that goes thru two points of the function
What is a secant line?
a line that just touches a function at a point.
What is a tangent line?
Folding the graph over the y axis, and having identical results on each side
What is graphical symmetry?
functions that are defined by different formulas
What is the definition of a piecewise function?
6
What is the degree of this polynomial?
9
What is the degree of this polynomial?
f(a+h)-f(a)/h
What is the difference quotient ?
[-4,4] [-2,3]
What is the domain and range of f(x)?
m = y2-y1/x2-x1
What is the equation for slope?
y-y1=m(x-x1)
What is the equaton for y intercept?
y=2(x-7)
What is the new equation of the translation?
lim f(x) =L x--->a
What is the short definition of a limit?
lim f(x) = f(a) x--->a
What is the short version of the Direct Substitution Property?
f(x) <= g(x) <= h(x)
What is the short version of the Squeeze Theorem?
y = mx+b
What is the slope intercept equation?
even
What is the symmetry of this graph?
change in position/change in time
What is velocity?
cosine graph
What type of graph is this?
exponential function
What type of graph is this?
logarithmic function
What type of graph is this?
root function
What type of graph is this?
sine graph
What type of graph is this?
1/2
What would the power function y =(2)^-1 give you?
When you can draw its graph with no skips, jumps, asymptotes, or holes
When is a function continuous?
when you cant separate limits into their individual pieces
When would you use the Squeeze Theorem?
-1
Where is the asymptote of this piecewise function?
5
Which translation corresponds to y=-f(x+4)
4
Which translation corresponds to y=1/3f(x)?
2
Which translation corresponds to y=2f(x+6)?
1
Which translation corresponds to y=f(x)+3
3
Which translation corresponds to y=f(x-4)?
Definition of a Function
relation between a set of inputs and outputs
5
s(t) = t^2 Estimate the instantaneous velocity at t= 2. [2,3]
4.1
s(t) =t^2 Estimate the instantaneous velocity at t = 2. [2,2.1]
4+h
s(t)=t^2 Estimate the instantaneous velocity at t = 2. [2, (2+h)]
6
s(t)=t^2 Estimate the instantaneous velocity at t=2 [2,4] HINT: use the slope equation
Domain
all values x is allowed to be
Range
all values y is allowed to be
3/4
valuate the limit, if it exists. (If it does not exist, enter NONE).