Unit 5 Review

IVT, MVT, EVT

100

What two conditions must be met for the Mean Value Theorem (MVT) to apply to a function f?

f is continuous on [a,b]/closed interval, differentiable on (a,b).

200

Which theorem guarantees a function f on a closed interval has both an absolute max and an absolute min. 

Extreme Value Theorem (EVT)

300

Rolle's Theorem is a special case of the Mean Value Theorem. What is the additional 3rd condition?

Sketch a graph where Rolle's Theorem would apply.

f(a) = f(b)

400

On [0,2], for f(x) = x², find the value of c the satisfies the MVT. 

A.K.A: Find the specific x-value where f'(c) = AROC.

c = 1

f'(c) = (4 - 0)/(2 - 0) = 2

f'(x) = 2x

f'(c) = 2 * 1 = 2

500

Why does MVT not apply to f(x) = |x| on the interval [-1,1]?

f(x) is not differentiable at x=0.

(Sharp Turn)

Find Critical Points/Points of Inflection

100

What is a Critical Point of a function f(x)?

f'(x) = 0 or f'(x) is undefined.

x must be in the domain of f.

200

Find the critical point(s) of f(x) = x²-6x+5.

Bonus:
Find the critical point(s) of the same f(x) on the interval [0,1].

x = 3, 2x - 6 = 0

Bonus: 0 and 1. No other extrema except for end points. 

300

Find the x-coordinate of the Point of Inflection for f(x) = x³-3x²

f''(x) = 6x-6

400

True/False: If f''(c) = 0, then x=c must be a point of inflection. 

False, f''(x) must also change sign. 

500

Find all of the critical points of f(x) = xe^x

f'(x) = e^x + xe^x = e^x(1+x).

x = -1

e^x is never = 0 or Undefined

Derivative Test (1st, Candidates, 2nd)

100

What must f'(x) do for a relative maximum to occur?

f goes from increasing to decreasing.

f' changes from + to -

200

If f'(c) = 0, and f''(c) > 0, what type of extrema, if any, is located at x=c?

Relative Minimum

300

Which test is used to find absolute extrema on a closed interval?

For this test, which points do you have to test in addition to critical points?

Candidates Test

-
Check Critical Points and End Points

400

If f'(x) = (x-2)(x+3), find the x-coordinate of the relative minimum.

x = 2

f' changes from  negative to positive at x = 2

Note: at x = -3, + to - sign change, relative max. 

500

If f'(2) = 0 and f''(x) = 1-2x,

Classify x = 2 using the 2nd Derivative Test. 

Relative Max

f"(2) = -3, Concave Down at the critical point. 

Curve Sketching

100

If f'(x) > 0 and f"(x) < 0, draw or describe the shape of f(x).

Increasing and Concave Down.

Upside-Down Parabola

200

The graph of f'(x) is a semicircle above the x-axis on the interval [-2,2]. On what interval is f(x) concave down?

(0,2) -> When f' is decreasing.

300

At what feature on the graph of f'(x) does f(x) have a point of inflection?

Relative extrema of f'(x).

Peaks, valleys, V.As.

400

The graph of f'(x) is a parabola opening downwards with a vertex at (3,4). At what x-value does f(x) have a point of inflection?

x=3. POIs occur at the relative extrema of f'.

500

Sketch f(x) for a function where f(0) = 0, f'(x) > 0, and f" changes from negative to positive at 0.

S-curve

Optimization & Implicit Relations

100

Write the objective function and constraint function for the following problem.

Find the maximum area for a fence with perimeter = 20ft

A = xy

P = 2x + 2y = 20

200

Find dy/dx for the implicit relation x²+y²=25

dy/dx = -x/y

300

What dimensions for a rectangle with Area = 100 minimizes the perimeter?

10x10 

Square

400

Find the slope of the tangent line to xy = 6 at (2,3)

m = f'(2,3) = -3/2

y + xdy/dx = 0 -> dy/dx = -y/x -> -3/2

500

Find d²y/dx² in terms of y only for y²=x

y' = 1/(2y)

y'' = -2y'/(4y^2) = -1/(4y^2 * y) = -1/(4y^3)