Midterm #1

Question 0

Odd Function

Answer 0

F(-x) = -F(x)

Symmetrical across the origin.

Question 1

Even Function

Answer 1

F(-x) = F(x)

Symmetrical across the y axis.

Question 2

Definition Of A Polynomial

Answer 2

Cannot have a non-integer exponent in a polynomial (i.e. no fractions, negative numbers, etc.)

Question 3

Definition Of A Power Function

Answer 3

A number raised to a constant power.

F(x) = x^-1 is a power function but not a polynomial.

Question 4

Rational Function

Answer 4

Quotient of 2 polynomials.

Question 5

Transformation - Translation - Shifts Along Vertical Plane

Answer 5

Y = F(x) + C is an upwards shift. 
Y = F(x) - C is a downwards shift.

Question 6

Transformation - Translation - Shifts Along Horizontal Plane

Answer 6

Y = F(X - C) is a rightwards shift. 
Y = F(X + C) is a leftwards shift.

Question 7

Transformation - Stretching - Stretch Along Vertical Plane

Answer 7

Y = F(X) x C is a stretch upwards.
Y = F(C) x 1/C is a shrink downwards.

Question 8

Transformation - Stretching - Stretch Along Horizontal Plane

Answer 8

Y = F(X/C) is a horizontal stretch. 
Y = F(XC) is a horizontal shrink.

Question 9

Transformation - Reflection - About X Axis

Answer 9

If y = f(x) is the original function,

y = -f(x) is a reflection about the x axis.

Question 10

Transformation - Reflection - About Y Axis

Answer 10

If y = f(x) is the original function,

y = f(-x) is a reflection about the y axis.

Question 11

Definition Of Left And Right Hand Limits

Answer 11

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.

Question 12

Proof Of A Limit

Answer 12

1. Both L and R limits exist

2. L = R

Question 13

Sin(x)/x

Answer 13

1

Question 14

Statement Of Squeeze Theorem

Answer 14

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.

Question 15

Definition Of Continuity

Answer 15

1. F(a) exists.
2. Lim as x goes to a exists
3. # 1 equals #2

Question 16

Theory Of Composition Of Continuous Functions

Answer 16

Show sin(x^2) is continuous at a = 0

F(x) = sin(x)
G(x) = x^2 
Sin(x^2) = F(G(x))

1. G(x) is continuous at x = a because G(a) = G(0^2) = 0
2. F(x) is continuous at G(a) because sin(0) = 0
3. Therefore sin(x^2) is cts. at a = 0.

Question 17

Proof Of Vertical Asymptote

Answer 17

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

Question 18

Horizontal Asymptotes

Answer 18

Show lim as x approaches either positive or negative infinity equal to some number.

Question 19

Limit Definition Of The Derivative Of A Function

Answer 19

F’(x) = Lim h->0 F(x + h) - f(x)/h

Question 20

Definition Of A Differentiable Function

Answer 20

When F’(a) exists; i.e. the limit function exists

Question 21

Differentiability and Continuity

Answer 21

If a function is different it is continuous.

HOWEVER, if a function is continuous, it may not be differentiable.

Question 22

Intermediate Value Theorem

Answer 22

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

Question 23

Showing Differentiability Of Absolute Value Functions

Answer 23

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.

Question 24

Normal Line

Answer 24

Perpendicular to the curve; negative reciprocal of the slope.

Question 25

Sine Values

Answer 25

Sin(0) = 0
Sin(pi/2) = 1
Sin(pi) = 0
Sin (3pi//) -1
Sin(2pi) = 0

Question 26

Cosine Values

Answer 26

Cos(0) = 1
Cos(pi/2) = 0
Cos(pi) = -1
Cos(3pi/2) = 0 
Cos(2pi) = 1

Question 27

Squeeze Theorem For “X”

Answer 27

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.