Question 1

Find the dimensions of the rectangle of largest area that can be inscribed in a semi-circle of radius R, assuming that one side of the rectangle lies on a diameter of the semi-circle.

Accepted Answer

The answer of Find the dimensions of the rectangle of...

Question 2

Find the intervals of increase and decrease for the function $f ( x ) = \sqrt { x ^ { 2 } + 1 }$ .

Accepted Answer

The answer of Find the intervals of increase and decrease...

Question 3

Find all critical numbers of the function $f ( x ) = x ^ { 5 } + 5 x ^ { 6 }$ .

Accepted Answer

The answer of Find all critical numbers of the function...

Question 4

The function $f ( x ) = x ^ { 2 } + \frac { 2 } { x }$ has a relative minimum at x = 1.

Accepted Answer

A) True 
 B)False

Question 5

A small manufacturing company estimates that the total cost in dollars of producing x radios per day is given by the formula $C = 0.1 x ^ { 2 } + 20 x + 500$ . Find the number of units that will minimize the average cost.

Accepted Answer

A)  100 
B)  147 
C)  36 
D)  71

Question 6

Determine the absolute maximum and minimum of $f ( x ) = x ^ { 4 } - 2 x ^ { 2 } + 5$ on the interval -2 $\le$ x $\le$1.

Accepted Answer

The answer of Determine the absolute maximum and minimum of...

Question 7

Find all the critical numbers of the function $f ( x ) = 2 x ^ { 2 } - 8 x + 7$ .

Accepted Answer

A)  -7 
B)  $- \frac { 7 } { 2 }$ 
C)  2 
D)  None

Question 8

Graph $f ( x ) = 2 x ^ { 3 } + 4 x ^ { 2 } - x$ .

Accepted Answer

The answer of Graph \(f ( x ) = 2...

Question 9

Locate all inflection points of $f ( x ) = x ^ { 4 } + 6 x ^ { 3 } - 24 x ^ { 2 } + 26$ .

Accepted Answer

A)  (1, 9) and (-4, -486) 
B)  (1, 9) 
C)  None 
D)  (0, 26)

Question 10

Find constants a, b, and c so that the graph of the function $f ( x ) = a x ^ { 2 } + b x + c$ has a relative maximum at (7, 34) and crosses the y-axis at (0, 6).

Accepted Answer

A)  $a = \frac { 4 } { 7 }$ b=8,c=6 
B)  $a = - \frac { 4 } { 7 }$ , b=-8, c=-6 
C)  $a = \frac { 4 } { 7 }$ , b=-8, c=6 
D)  $a = - \frac { 4 } { 7 }$ , b=8, c= 6

Question 11

Find all vertical and horizontal asymptotes of the graph of the given function. $$f ( x ) = \frac { x ^ { 2 } - 3 x } { x ^ { 2 } + 5 x + 6 }$$

Accepted Answer

A)  Vertical asymptotes: x = -3, x = -2; horizontal asymptote: y = 1 
B)  Vertical asymptotes: x = -3, x = -2; horizontal asymptote: y = 3 
C)  Vertical asymptotes: x = -3, x = -2, x = 3; horizontal asymptote: y = 0 
D)  Vertical asymptotes: x = -3, x = -2, x = 3; horizontal asymptote: y = 1

Question 12

Find A and B so that the graph of $f ( x ) = \frac { 29 - A x } { B x + 15 }$ has y = 11 as a horizontal asymptote and x = 2 as a vertical asymptote.

Accepted Answer

The answer of Find A and B so that the...

Question 13

Find all critical points of $f ( x ) = 3 x ^ { 4 } - 2 x ^ { 3 } - 12 x ^ { 2 } + 18 x$ , and use the second derivative test to classify each as a relative maximum, a relative minimum, or neither.

Accepted Answer

The answer of Find all critical points of \(f (...

Question 14

Graph $f ( x ) = 1.2 - 0.6 x + 0.2 x ^ { 2 }$ .

Accepted Answer

The answer of Graph \(f ( x ) = 1.2...

Exam 3: Additional Applications of the Derivative

Find the dimensions of the rectangle of largest area that can be inscribed in a semi-circle of radius R, assuming that one side of the rectangle lies on a diameter of the semi-circle.

Find the intervals of increase and decrease for the function $f ( x ) = \sqrt { x ^ { 2 } + 1 }$ .

Find all critical numbers of the function $f ( x ) = x ^ { 5 } + 5 x ^ { 6 }$ .

The function $f ( x ) = x ^ { 2 } + \frac { 2 } { x }$ has a relative minimum at x = 1.

A small manufacturing company estimates that the total cost in dollars of producing x radios per day is given by the formula $C = 0.1 x ^ { 2 } + 20 x + 500$ . Find the number of units that will minimize the average cost.

Determine the absolute maximum and minimum of $f ( x ) = x ^ { 4 } - 2 x ^ { 2 } + 5$ on the interval -2 $\le$ x $\le$ 1.

Find all the critical numbers of the function $f ( x ) = 2 x ^ { 2 } - 8 x + 7$ .

Graph $f ( x ) = 2 x ^ { 3 } + 4 x ^ { 2 } - x$ .

Locate all inflection points of $f ( x ) = x ^ { 4 } + 6 x ^ { 3 } - 24 x ^ { 2 } + 26$ .

Find constants a, b, and c so that the graph of the function $f ( x ) = a x ^ { 2 } + b x + c$ has a relative maximum at (7, 34) and crosses the y-axis at (0, 6) .

Find all vertical and horizontal asymptotes of the graph of the given function. $f ( x ) = \frac { x ^ { 2 } - 3 x } { x ^ { 2 } + 5 x + 6 }$

Find A and B so that the graph of $f ( x ) = \frac { 29 - A x } { B x + 15 }$ has y = 11 as a horizontal asymptote and x = 2 as a vertical asymptote.

Find all critical points of $f ( x ) = 3 x ^ { 4 } - 2 x ^ { 3 } - 12 x ^ { 2 } + 18 x$ , and use the second derivative test to classify each as a relative maximum, a relative minimum, or neither.

Graph $f ( x ) = 1.2 - 0.6 x + 0.2 x ^ { 2 }$ .

Exam 3: Additional Applications of the Derivative

Find the dimensions of the rectangle of largest area that can be inscribed in a semi-circle of radius R, assuming that one side of the rectangle lies on a diameter of the semi-circle.

Find the intervals of increase and decrease for the function f(x)=x2+1f ( x ) = \sqrt { x ^ { 2 } + 1 }f(x)=x2+1​ .

Find all critical numbers of the function f(x)=x5+5x6f ( x ) = x ^ { 5 } + 5 x ^ { 6 }f(x)=x5+5x6 .

The function f(x)=x2+2xf ( x ) = x ^ { 2 } + \frac { 2 } { x }f(x)=x2+x2​ has a relative minimum at x = 1.

A small manufacturing company estimates that the total cost in dollars of producing x radios per day is given by the formula C=0.1x2+20x+500C = 0.1 x ^ { 2 } + 20 x + 500C=0.1x2+20x+500 . Find the number of units that will minimize the average cost.

Determine the absolute maximum and minimum of f(x)=x4−2x2+5f ( x ) = x ^ { 4 } - 2 x ^ { 2 } + 5f(x)=x4−2x2+5 on the interval -2 ≤\le≤ x ≤\le≤ 1.

Find all the critical numbers of the function f(x) =2x2−8x+7f ( x ) = 2 x ^ { 2 } - 8 x + 7f(x) =2x2−8x+7 .

Graph f(x)=2x3+4x2−xf ( x ) = 2 x ^ { 3 } + 4 x ^ { 2 } - xf(x)=2x3+4x2−x .

Locate all inflection points of f(x) =x4+6x3−24x2+26f ( x ) = x ^ { 4 } + 6 x ^ { 3 } - 24 x ^ { 2 } + 26f(x) =x4+6x3−24x2+26 .

Find constants a, b, and c so that the graph of the function f(x) =ax2+bx+cf ( x ) = a x ^ { 2 } + b x + cf(x) =ax2+bx+c has a relative maximum at (7, 34) and crosses the y-axis at (0, 6) .

Find all vertical and horizontal asymptotes of the graph of the given function. f(x) =x2−3xx2+5x+6f ( x ) = \frac { x ^ { 2 } - 3 x } { x ^ { 2 } + 5 x + 6 }f(x) =x2+5x+6x2−3x​

Find A and B so that the graph of f(x)=29−AxBx+15f ( x ) = \frac { 29 - A x } { B x + 15 }f(x)=Bx+1529−Ax​ has y = 11 as a horizontal asymptote and x = 2 as a vertical asymptote.

Find all critical points of f(x)=3x4−2x3−12x2+18xf ( x ) = 3 x ^ { 4 } - 2 x ^ { 3 } - 12 x ^ { 2 } + 18 xf(x)=3x4−2x3−12x2+18x , and use the second derivative test to classify each as a relative maximum, a relative minimum, or neither.

Graph f(x)=1.2−0.6x+0.2x2f ( x ) = 1.2 - 0.6 x + 0.2 x ^ { 2 }f(x)=1.2−0.6x+0.2x2 .

Find the intervals of increase and decrease for the function $f ( x ) = \sqrt { x ^ { 2 } + 1 }$ .

Find all critical numbers of the function $f ( x ) = x ^ { 5 } + 5 x ^ { 6 }$ .

The function $f ( x ) = x ^ { 2 } + \frac { 2 } { x }$ has a relative minimum at x = 1.

A small manufacturing company estimates that the total cost in dollars of producing x radios per day is given by the formula $C = 0.1 x ^ { 2 } + 20 x + 500$ . Find the number of units that will minimize the average cost.

Determine the absolute maximum and minimum of $f ( x ) = x ^ { 4 } - 2 x ^ { 2 } + 5$ on the interval -2 $\le$ x $\le$ 1.

Find all the critical numbers of the function $f ( x ) = 2 x ^ { 2 } - 8 x + 7$ .

Graph $f ( x ) = 2 x ^ { 3 } + 4 x ^ { 2 } - x$ .

Locate all inflection points of $f ( x ) = x ^ { 4 } + 6 x ^ { 3 } - 24 x ^ { 2 } + 26$ .

Find constants a, b, and c so that the graph of the function $f ( x ) = a x ^ { 2 } + b x + c$ has a relative maximum at (7, 34) and crosses the y-axis at (0, 6) .

Find all vertical and horizontal asymptotes of the graph of the given function. $f ( x ) = \frac { x ^ { 2 } - 3 x } { x ^ { 2 } + 5 x + 6 }$

Find A and B so that the graph of $f ( x ) = \frac { 29 - A x } { B x + 15 }$ has y = 11 as a horizontal asymptote and x = 2 as a vertical asymptote.

Find all critical points of $f ( x ) = 3 x ^ { 4 } - 2 x ^ { 3 } - 12 x ^ { 2 } + 18 x$ , and use the second derivative test to classify each as a relative maximum, a relative minimum, or neither.

Graph $f ( x ) = 1.2 - 0.6 x + 0.2 x ^ { 2 }$ .