Midterm #1 Flashcards
Odd Function
F(-x) = -F(x)
Symmetrical across the origin.
Even Function
F(-x) = F(x)
Symmetrical across the y axis.
Definition Of A Polynomial
Cannot have a non-integer exponent in a polynomial (i.e. no fractions, negative numbers, etc.)
Definition Of A Power Function
A number raised to a constant power.
F(x) = x^-1 is a power function but not a polynomial.
Rational Function
Quotient of 2 polynomials.
Transformation - Translation - Shifts Along Vertical Plane
Y = F(x) + C is an upwards shift. Y = F(x) - C is a downwards shift.
Transformation - Translation - Shifts Along Horizontal Plane
Y = F(X - C) is a rightwards shift. Y = F(X + C) is a leftwards shift.
Transformation - Stretching - Stretch Along Vertical Plane
Y = F(X) x C is a stretch upwards. Y = F(C) x 1/C is a shrink downwards.
Transformation - Stretching - Stretch Along Horizontal Plane
Y = F(X/C) is a horizontal stretch. Y = F(XC) is a horizontal shrink.
Transformation - Reflection - About X Axis
If y = f(x) is the original function,
y = -f(x) is a reflection about the x axis.
Transformation - Reflection - About Y Axis
If y = f(x) is the original function,
y = f(-x) is a reflection about the y axis.
Definition Of Left And Right Hand Limits
The real number L is the left hand/right hand limit of f(x) at x = a if f(x) approaches L when x approaches a and x < a/x > a.
Proof Of A Limit
- Both L and R limits exist
2. L = R
Sin(x)/x
1
Statement Of Squeeze Theorem
By the squeeze theorem, because f(x) < or = g(x) < or = h(x) and lim as x goes to 0 of f(x) equals “ of h(x) equals 0; “ g(x) also equals 0.
Definition Of Continuity
- F(a) exists.
- Lim as x goes to a exists
- # 1 equals #2
Theory Of Composition Of Continuous Functions
Show sin(x^2) is continuous at a = 0
F(x) = sin(x) G(x) = x^2 Sin(x^2) = F(G(x))
- G(x) is continuous at x = a because G(a) = G(0^2) = 0
- F(x) is continuous at G(a) because sin(0) = 0
- Therefore sin(x^2) is cts. at a = 0.
Proof Of Vertical Asymptote
Show that either the left or right limit as x approaches some number is equal to infinity.
eg.
F(x) = 1/(x^2 - 1)
F(x) = 1(x + 1)(x - 1)
Lim as x -> 1- = 1(1 + 1)(1 - 1) = 1/(2)(0)
But, 1- means a number very small close to 1 from the left, so .9999999 which would be .999999 - 1 which is a negative number so 1/a very small negative number = negative infinity
Horizontal Asymptotes
Show lim as x approaches either positive or negative infinity equal to some number.
Limit Definition Of The Derivative Of A Function
F’(x) = Lim h->0 F(x + h) - f(x)/h
Definition Of A Differentiable Function
When F’(a) exists; i.e. the limit function exists
Differentiability and Continuity
If a function is different it is continuous.
HOWEVER, if a function is continuous, it may not be differentiable.
Intermediate Value Theorem
F(x) is continuous because (i.e. all polynomials are continuous), therefore because f(some given number) < number given < f(some other given number), there exists some x value such that f(x) = number given
Showing Differentiability Of Absolute Value Functions
If the point a is the point of intersection of the two piecewise functions that make up an absolute value function (where the abs goes to 0), then use the different functions in the limit differentiation equation for the left and right limits. If the point a is not that point, show that one limit is positive and the other is negative.
Normal Line
Perpendicular to the curve; negative reciprocal of the slope.
Sine Values
Sin(0) = 0 Sin(pi/2) = 1 Sin(pi) = 0 Sin (3pi//) -1 Sin(2pi) = 0
Cosine Values
Cos(0) = 1 Cos(pi/2) = 0 Cos(pi) = -1 Cos(3pi/2) = 0 Cos(2pi) = 1
Squeeze Theorem For “X”
Must show left and right limits of the outside functions. For left limit assume x is negative and flip inequalities. Therefore left and right limit of squeezed function also equal to that limit.
