Applications of Derivatives Review

An equation of the line tangent to the graph of $y=x^3+3x^2+2$ at its point of inflection is 

•  $y=-3x+1$  

•  $y\ =-3x-7$  

•  $y=x+5$  

•  $y=3x+1$  

Which of the following theorems may be applied to the function $y=\left|x-3\right|+b,\ \ b>0,$ over the interval [2, 4]?

I.      Mean Value Theorem

II.    Intermediate Value Theorem

III.   Extreme Value Theorem 

• I only

• II only

• III only

• II and III only

• I, II, and III

Tags

CCSS.HSF.IF.B.5

The derivative of the function g is $g'\left(x\right)=\cos\left(\sin x\right).$ At the point where  $x=0$ the graph of g
I.  is increasing
II.  is concave down
III.  attains a relative maximum point

• I only

• II only

• I and III only

• I, II, III

Tags

CCSS.HSF.IF.B.4

CCSS.HSF.IF.C.7

Let f be a differentiable function with values at selected inputs recorded in the table above. Which of the following must be true for $-1<x<4?$  

• The graph of f is concave up.

• The graph of f has at least two points of inflection.

• f is increasing.

• f has at least two relative extrema.

Tags

CCSS.HSF.IF.B.4

CCSS.HSF.IF.C.7

CCSS.HSF.IF.B.5

Water is flowing into a spherical tank with 6 foot radius at the constant rate of  $30\pi$ cubic feet per hour.  When the water is h feet deep, the volume of water in the tank is given by $V=\frac{\pi h^2}{3}\left(18-h\right).$  What is the rate at which the depth of the water in the tank is increasing at the moment when the water is 2 feet deep?

• 0.5 ft per hr

• 1.0 ft per hr

• 1.5 ft per hr

• 2.0 ft per hr

Tags

CCSS.HSA.SSE.A.1

CCSS.HSA.CED.A.1

CCSS.HSA.REI.B.4

The graph of

the derivative of f, is shown.  The graph of  $f'$  has a horizontal tangent at  $x=0.$  Which of the following statements are true about the function f?

I. f is increasing on the interval (-2, -1).
II.  f has an inflection point at  $x=0.$  
III.  f is concave up on the interval (-1, 0).

• I only

• II only

• III only

• I and II only

If $f\left(x\right)=x^3-5x^2+3x,$ then the graph of f is decreasing and concave down on the interval

•  $\left(0,\ \frac{1}{3}\right)$  

•  $\left(\frac{1}{3},\ \frac{2}{3}\right)$  

•  $\left(\frac{1}{3},\ \frac{5}{3}\right)$  

•  $\left(\frac{5}{3},\ 3\right)$  

Tags

CCSS.HSF.IF.B.4

CCSS.HSF.IF.C.7

• M

• N

• P

• R

Tags

CCSS.HSA.APR.B.3

Let $f\left(x\right)=x^4+ax^2+b,$ where a and b are constants.  The graph of f has a relative maximum at (0, 1) and an inflection point when  $x=1.$ The values of a and b are

• a = 1  and  b = -6

• a = 1  and  b = 6

• a = -6  and  b = 5

• a = -6  and  b = 1

Tags

CCSS.HSA.SSE.A.1

CCSS.HSA.APR.A.1

CCSS.HSA.REI.D.10

The function f is continuous on the closed interval [1, 5]. The table below gives selected values of f on this interval. Which of the following statements must be TRUE?

• $f'\left(x\right)>0$ for $1<x<3$

• $f''\left(x\right)<0$ for $3<x<5$

• The minimum value of f on [1, 5] must be -2

• There exists a number c, $1<c<5$ for which $f\left(c\right)=0$

Tags

CCSS.HSA.APR.B.3