Calc. 1 Midterm 1

Question 1

Function

Answer 1

Each input (x) is only assigned to one output (y)

Question 2

Function Representation

Answer 2

Graphically, Algebraically, Verbally, & Numerically

Question 3

Explain when it is a vertical/horizontal stretch/shrink, a vertical/horizontal shift, and when it flips over the x/y axis

Answer 3

Vertical Stretch: y = 3f(x)
Vertical Shrink: y = .5f(x)
Horizontal Stretch: y = f(.5x)
Horizontal Shrink: y = f(3x)
Vertical Shift (up): y = f(x) + 5
Vertical Shift (down): y = f(x) - 5
Horizontal Shift (left): y = f(x+15)
Horizontal Shift (right): y = f(x-15)
X-Axis Flip: -f(x)
Y-Axis Flip: f(-x)

Question 4

based on f((.5x)-3) explain the steps in order…do the same for f(.5(x-3))…and why the difference?

Answer 4

f((.5x)-3)
- shift right by 3
- horizontal stretch by factor of 2
f(.5(x-3))
- horizontal stretch by factor of 2
- shift right by 3
Why different?
- Apply outermost parenthesis first and work inside

Question 5

explain how to solve g(f(x))

Answer 5

• evaluate f(x) at x
• with F(x) output evaluate g(x)

Question 6

Explain the proper composition of the word problem:
m(t): # tennis balls manufactured in 2000+t
c(n): cost to manufacture n tennis balls ($)

Answer 6

c(m(t)) = cost to manufacture m(t) tennis balls in 200+t

m(c(n)) = bruh don’t be dumb

Question 7

angle is measured based on?

Answer 7

arc length

Question 8

sin^2(x) + cos^2(x) =

Answer 8

1

Question 9

tan (x) = and what that means

Answer 9

sin(x)/cos(x) which is the slope of those lines

Question 10

Domain of cos(x)

Answer 10

all real #

Question 11

Domain of sin(x)

Answer 11

all real #

Question 12

Domain of tan(x)

Answer 12

all real #’s except where cos(x) = 0

Question 13

graph cos(x) =

Answer 13

graph sin(x) + pi/2

Question 14

equation for finding sin or cos function

Answer 14

Asin(B(t-h)) + K

Question 15

Define each term in Asin(B(t-h)) + K

Answer 15

A: Amplitude (half of curve height)
sin: if it stars from corner
cos: if it starts higher up
B: (2pi)/period
Period: from max to max (cos) or from midline to 2nd midline away
t = variable
h = shift of max (cos) or middle (sin) from y-axis
K : from x-axis to midline of curve

Question 16

Typical exponential model form and explanation

Answer 16

e(t) = ab^t
e(t) = 100(1.3)^t
a = initial value
b = growth factor
*b>1 = growth
*b<1 = decay

Question 17

Inverse Function

Answer 17

1-1 if no two inputs have the same outputs and must pass the horizontal line test

Question 18

Horizontal Line Test

Answer 18

Determines if inverse function is a one-one if the line meets the graph at only 1 point

Question 19

Domain f(x) = ________ inverse f(x)
Range f(x) = __________ inverse f(x)

Answer 19

Domain f(x) = Range inverse f(x)
Range f(x) = Domain inverse f(x)

Question 20

Logarithms

Answer 20

for any b>0, the function b^x is one-one & the inverse of the logarithum is log(base b)(x)
ex. 3^2 = 9 is same as log(base 3) (9) = 2

Question 21

Domain & range of inverse sin

Answer 21

Domain: [-pi/2, pi/2]
Range: [-1, 1]

Question 22

Domain & range of inverse cos

Answer 22

Domain: [0, pi]
Range: [-1, 1]

Question 23

Domain & range of inverse tan

Answer 23

Domain: (- infinity, infinity)
Range: (-pi/2, pi/2)

Question 24

Continuous Function F
Formally:
Informally:

Answer 24

Formally: continuous @a if the limit as x approaches a f(x) = f(a)
Informally: no break in the graph @ x=a

Question 25

Being continuous at x = a

Answer 25

1) f(a) is defined
2) limit of f(x) as x approaches a has to exist
3) the two are equal (if x=a isn’t in the domain (a isn’t defined) then then f(x) isn’t continuous

Question 26

Ways of finding limits

Answer 26

1. Direct substitution
2. Algebraically
3. Squeeze Theorem
4. Leading Coefficients