Unit 10 Math Test

Question 1

Extrema happens in two places

Answer 1

1. Turning Points
2. Endpoints

Question 2

Absolute (global) maximum f(c)

Answer 2

f(x) ≤ f(c) for all x values on the domain

Question 3

Absolute (global) minimum f(c)

Answer 3

f(x) ≥ f(c) for all x values on the domain

Question 4

Relative (local) maximum f(c)

Answer 4

f(x) ≤ f(c) for all x values in an open interval containing c

Question 5

Relative (local) minimum f(c)

Answer 5

f(x) ≥ f(c) for all x values in an open interval containing c

Question 6

Extreme Value Theorem

Answer 6

If f is continuous on a closed interval [a, b], then f has both a maximum and minimum value on the interval

Question 7

Critical Point

Answer 7

A point in the domain which f’ = 0 or f’ does not exist

Question 8

Finding absolute extrema

Answer 8

1. Determine the critical points
   a. take the derivate
   b. set the derivate to zero and solve
   c. determine any places where the derivative is undefined
2. Test the critical points and end points. The largest value is the absolute max and the smallest is the absolute min

Question 9

Relationship between increasing/decreasing functions and derivative

Answer 9

f’ > 0 at each point in (a,b) then f increases on (a,b)

and opposite

Question 10

Local Max and Local Min

Answer 10

A function f has a local maximum at x = c if f’ changes from positive to negative

A function f has a local minimum at x = c if f’ changes from negative to positive

Question 11

Are endpoints local extrema? [a, b]

Answer 11

The left endpoint “a”
1. a local maximum if f’ is negative for x>a
2. a local minimum if f’ is positive for x>a

The right endpoint “b”
1. a local maximum if f’ is positive for x<b
2. a local minimum if f’ is negative for x<b

Question 12

Method to finding all extrema of a function

Answer 12

Number line

Question 13

Concave up is a smile

Answer 13

Concave down is a frown

Question 14

Inflection points

Answer 14

Points when curves change concavity
f’’ = 0 or does not exist

Question 15

Definition of Concavity

Answer 15

f(x) is concave up on an interval if all the tangents to the curve are below the graph

f(x) is concave down on an interval if all the tangents to the curve are above the graph

Question 16

Concavity: f’ is increasing (THINK SIGN AND NUMBER)

Answer 16

Concave up

Question 17

Concavity: f’ is decreasing (THINK SIGN AND NUMBER)

Answer 17

Concave down

Question 18

Velocity is the

Answer 18

First derivative
Change of position over time d(p)/d(t)

Question 19

Acceleration is the

Answer 19

Second derivative
Change of velocity over time d(v)/d(t)

Question 20

If velocity and acceleration have the same sign

Answer 20

speeding up

Question 21

If velocity and acceleration have the opposite sign

Answer 21

slowing down

Question 22

Five things to remember for word problems

Answer 22

1. List needed formulas
2. Define variables
3. Draw pictures
4. Write the equations using variables
5. Write the full sentence for the answer

Question 23

Area of a triangle

Answer 23

A = 1/2 bh

Question 24

Area of a triangle with trig

Answer 24

A = 1/2 ab sin(θ)

Question 25

Tangent line approximation

Answer 25

f(x) ≈ f(a) + f’(a)(x-a)

Question 26

Error

Answer 26

E(x) = f(x) - f(a) - f’(a)(x-a)

Question 27

Mean Value Theorem

Answer 27

If f is a continuous function on the closed interval [a,b] and differentiable on the open interval (a,b) there exist a number c in (a,b) such that

f’(c) = f(b) - f(a)/b-a

Question 28

Area of a circle

Answer 28

A = πr^2

Question 29

Circumference of a circle

Answer 29

2πr