Less3 Trig Review

Question 1

What is calculus?

Answer 1

the mathematics of change:

average speed,

instantaneous speed,

acceleration

Question 2

What is the tangent line problem?

Answer 2

This is the heart of

differential calculus

^{solved by Newton and Leibniz independently}

the slope of any tangent line is the rate of change

Question 3

What is slope?

Answer 3

^rise / _run

the vertical change over the horizontal change

As the x value changes, how much does the dependent variable y change?

slope = m

Question 4

How did Newton and Leibniz solve the tangent line problem?

Answer 4

The calculated the slope of 2 points on a line/curve.

^{This is the secant line = a line that connects 2 points on a curve. (This is like the average speed of a vehicle over a distance)}

They then calculated the slope of the line as the second point got closer to the first.

Those slopes tended toward the slope of the tangent line.

This is “taking the limit.”

Question 5

With y = x², how can you use algebra to find the slope at any point?

Answer 5

1. Find 2 points on the line (2,4) and (3,9)
2. ^{assume 2,4 is the base point}
3. The rise over the run is:
4. ^{x² - 4} / _{x - 2}
5. ^{(x - 2)(x + 2)}/_{(x - 2)}
6. The slope at any given point is x + 2

Question 6

How do you notate limits?

Answer 6

lim_x→2 (x² - 4)/_{(x - 2)}

= lim_x→2(x + 2) = 4

Question 7

What is the point slope formula and how do you use it to find slope?

Answer 7

(y₂ - y₁) = m(x₂ - x₁)

rise = slope*run

[^rise/_run = slope]

y - 4 = 4(x - 2)

y = 4x - 4

at x = 2, y = 4

Question 8

Find the equation of the tangent line to the parabola y = x² at the point (3,9)

Answer 8

m = x² - 9/x - 3

m = ^{(x - 3)(x + 3)}/_{(x - 3)}

m = x + 3, x not = -3

m = 6

(y - 9) = 6(x - 3)

y = 6x - 9

Question 9

What is the domain of

f(x) = ¹/_{(x - 1)}?

Answer 9

AR#s except 1

Question 10

What is true about all outcomes of:

f(x) = |x|

In other words, what is its range?

Answer 10

f(x) is always positive

range ≥ 0

Question 11

What is a one-to one function? What test can you use?

Give an example.

Answer 11

For each y-value, there is only one x-value.

Horizontal line test.

f(x) = x²

Question 12

Is f(x) = x³ a

one-to-one function?

Answer 12

Yes.

Question 13

What is the vertical line test for?

Answer 13

It tells whether it’s a function at all.

Question 14

What’s an even function?

Answer 14

f(-x) = f(x)

Symmetric about the y-axis.

Polynomials to an even power.

Not one-to-one.

Question 15

What’s an odd function?

Answer 15

f(-x) = - f(x)

Symmetric about the origin.

Polynomials to an odd power.

One-to-one function

Question 16

How does the graph of the following function shift from f(x) = √x?

f(x) = √(x - 3) +5

Answer 16

3 to the right, up 5

x cannot be less than 3

the first point is (3,5)

Question 17

What are the 2 viewpoints of trigonometry?

Answer 17

The triangle viewpoint and the unit-circle viewpoint

Question 18

What is the unit characteristic of the unit-circle method?

Answer 18

The x-value is the cos and the y-value is the sin

Question 19

How does this lead to the most fundamental trig identity?

Answer 19

By the Pythagorean Theorem,

sin² + cos² = 1

Question 20

What does the sin function look like?

Answer 20

attached

Question 21

What does the cosine function look like?

Answer 21

attached

Question 22

What are the asymptotes of the tangent graph?

Answer 22

When the tangent line is flat (cos in the denominator is 0)

π/2 ± π

Question 23

Example of trig equation:

Find all values of Θ where

sinΘ = ½

Answer 23

^π/₆ + n2π and ^5π/₆ + n2π

Question 24

What are the domain and range of

f(x) = x² - 5?

Answer 24

domain: AR#s
range: ≥ -5

Question 25

What are the domain and range of:

f(x) = -√(x + 3)?

Answer 25

Domain: ≥ -3

Range: negative numbers

Question 26

What are the domain and range of:

f(x) = cot(x)?

Answer 26

cot(x) = cosx/sinx

domain: sinx not = 0 (^π/₂ and ^3π/₂
range: AR#s

Question 27

Is ³√x an odd or even function?

Answer 27

Odd because ³√-x = -y

³√-8 = -2