[Notes & HW Answers] 4.1 Using First and Second Derivatives

[Notes & HW Answers] 4.1 Using First and Second Derivatives.pdf

3.14MB

 

[Prepwork 4.1]

Q1. Suppose f(x)f(x) is a continuous function that has a local maximum at the point (3,2).Which of the following statements must be true?

A: (1) If a is just a little greater than 3, then f(a) ≤ 2. 

(2) f(x) has a critical point at x=3.

(3)  If a is just a little less than 3, then f(a) ≤ 2.

Q2. Suppose f(x)f(x) is a continuous function that has a local minimum at (1,6)(1,6) and no other critical points on 0<x<30<x<3. For each of the following statements, decide whether the statement is always true, sometimes true, or never true.

A:

f(x)is increasing for0<x<1and decreasing for 1<x<3? Never true 
f(x) is decreasing for 0<x<1 and increasing for 1<x<3? Always true
f′(x) is positive for 0<x<1 and negative for 1<x<3? Never true 
f′(x) is negative for 0<x<1 and positive for 1<x<3? Always true 
f″(1)>0  Sometimes true
f″(1)<0 Never true 
f″(1)=0  Sometimes true

Q3. Suppose f(x) is a twice-differentiable function with f′′(x)=e^xx^2(x^2+x−6). At what values of x does f(x) have an inflection point?

A: at x= -3, 2

[HW 4.1]

Q1. Use a graph below of f(x)=3e^(−7x^2) to estimate the x-values of any critical points and inflection points of f(x).

A: 

critical points: x= 0
inflection points: x= −0.3, 0.3

Next, use derivatives to find the xx-values of any critical points and inflection points exactly.
critical points: x= 0
inflection points: x= (42/588)^(1/2),  -(42/588)^(1/2).

Q2. The following table gives values of the differentiable function y=f(x).

x	0	1	2	3	4	5	6	7	8	9	10
y	-3	-4	-5	-2	1	3	1	-1	-2	1	3

Estimate the x-values of critical points of f(x) on the interval 0<x<10. Classify each critical point as a local maximum, local minimum, or neither.
(Enter your critical points as comma-separated xvalue,classification pairs. For example, if you found the critical points x=−2 and x=3, and that the first was a local minimum and the second neither a minimum nor a maximum, you should enter (-2,min), (3,neither). Enter none if there are no critical points.)

A:

critical points and classifications: (2,min),(5,max),(8,min)

Now assume that the table gives values of the continuous function y=f′(x) (instead of f(x)). Estimate and classify critical points of the function f(x).

critical points and classifications: (31/8,min),(34/5,max),(44/5,min)

Q3. Find and classify the critical points off(x)=5x5(1−x)4f(x)=5x5(1−x)4 as local maxima and minima.

A:

Critical points: x= 0,5/9​,1
Classifications: neither,max,min
(Enter your critical points and classifications as comma-separated lists, and enter the types in the same order as your critical points. Note that you must enter something in both blanks for either to be evaluated. For the types, enter min, max, or neither.

 

Q4.The following shows graphs of three functions, A (in black), B (in blue), and C (in green). If these are the graphs of three functions ff, f′f′, and f′′f″, identify which is which.

====image ===

A: f=B;f′=C;f″= A