Question 1

You measure the side of a cube to be 14 centimeters long and conclude that the volume of the cube is $14 ^ { 3 } = 2,744$ cubic centimeters. If your measurement of the side is accurate to within 5%, approximately how accurate is your calculation of this volume? Round to two decimal places, if necessary. A) Maximum error in volume is about ±29.4 cm³ B) Maximum error in volume is about ±411.6 cm³ C) Maximum error in volume is about ±2.1 cm³ D) Maximum error in volume is about ±5,762.4 cm³

Accepted Answer

The maximum error in the measurement of the side of the cube is 5% of 14 cm, which is 0.7 cm. The error in volume calculation due to this error in measurement can be approximated by considering the derivative of the volume with respect to the side length, $V = s^3$, which gives $dV = 3s^2 ds$. Substituting $s = 14$ cm and $ds = 0.7$ cm, we get $dV = 3(14)^2(0.7) \approx 411.6$ cm³. This represents the maximum error in the volume calculation, which is about ±411.6 cm³.

Question 2

If the total cost of manufacturing q units of a certain commodity is C(q) = (3q + 1)(5q + 7), use marginal analysis to estimate the cost of producing the 19th unit, in dollars.

Accepted Answer

The answer of If the total cost of manufacturing q...

Question 3

An efficiency study of the morning shift at a certain factory indicates that an average worker arriving on the job at 8:00 A.M. will have assembled $f ( x ) = - x ^ { 3 } + 10 x ^ { 2 } - 3 x$ transistor radios x hours later. Approximately how many radios will the worker assemble between 10:00 and 10:15 A.M.?
A) Approximately 25 radios
B) Approximately 375 radios
C) Approximately 6 radios
D) Approximately 26 radios

Accepted Answer

To find the number of radios assembled between 10:00 and 10:15 A.M., calculate the difference in $f(x)$ at $x = 2$ (10:00 A.M.) and $x = 2.25$ (10:15 A.M.). $f(2) = -2^3 + 10(2)^2 - 3(2) = 26$ and $f(2.25) = -(2.25)^3 + 10(2.25)^2 - 3(2.25) \approx 32$. The difference is approximately 6 radios.

Question 4

If $x ^ { 3 } + y ^ { 3 } = x + y$ , then $\frac { d y } { d x } = \frac { 3 x ^ { 2 } - 1 } { 3 y ^ { 2 } - 1 }$ .

Accepted Answer

The answer of If \(x ^ { 3 } +...

Question 5

Find $\frac { d y } { d x }$ , where $x y ^ { 3 } - 3 x ^ { 2 } = 7 y$ .
A) $y ^ { 3 } - 6 x - 7$
B) $\frac { 6 x - y ^ { 3 } } { 3 x y ^ { 2 } - 7 }$
C) $y ^ { 3 } - 6 x$
D) $\frac { 6 x ^ { 2 } } { y ^ { 3 } }$

Accepted Answer

To find $\frac { d y } { d x }$, we differentiate both sides of the equation $x y ^ { 3 } - 3 x ^ { 2 } = 7 y$ with respect to $x$, using the product rule and chain rule where necessary. Differentiating $x y ^ { 3 }$ gives $y^3 + 3xy^2 \frac{dy}{dx}$, and differentiating $-3x^2$ gives $-6x$. On the right side, differentiating $7y$ gives $7\frac{dy}{dx}$. Setting these equal gives $y^3 + 3xy^2 \frac{dy}{dx} - 6x = 7\frac{dy}{dx}$. Solving for $\frac{dy}{dx}$ gives $\frac{dy}{dx} = \frac{6x - y^3}{3xy^2 - 7}$, which matches choice B.

Question 6

Find $\frac { d y } { d x }$ , where $\sqrt { x } + \sqrt { y } = x y$ .

Accepted Answer

The answer of Find \(\frac { d y } {...

Question 7

Find $\frac { d y } { d x }$ , where $\frac { 3 } { x } + \frac { 1 } { 2 y } = 5$ .

Accepted Answer

The answer of Find \(\frac { d y } {...

Question 8

If $x ^ { 2 } + 3 x y + y ^ { 2 } = 15$ , then $\frac { d y } { d x } = 2 x + 3 y$ .

Accepted Answer

To find $\frac{dy}{dx}$, we need to differentiate both sides of the equation $x^2 + 3xy + y^2 = 15$ with respect to $x$, applying the product rule to $3xy$. Differentiating $x^2$ gives $2x$, for $3xy$ it gives $3y + 3x\frac{dy}{dx}$ (using the product rule), and for $y^2$ it gives $2y\frac{dy}{dx}$ (since $y$ is a function of $x$). Setting the derivative of the constant $15$ to $0$, we get $2x + 3y + 3x\frac{dy}{dx} + 2y\frac{dy}{dx} = 0$, which does not simplify to $\frac{dy}{dx} = 2x + 3y$.

Question 9

If $x ^ { 2 } y + x y ^ { 2 } = 7$ , then $\frac { d y } { d x } = 2 x y + y ^ { 2 }$ .

Accepted Answer

To find $\frac{dy}{dx}$, we need to differentiate the given equation $x^2y + xy^2 = 7$ implicitly with respect to $x$. Differentiating both sides with respect to $x$, we get $2xy + x^2\frac{dy}{dx} + y^2 + 2xy\frac{dy}{dx} = 0$. This does not simplify directly to $\frac{dy}{dx} = 2xy + y^2$, indicating the statement is false.

Question 10

If $x ^ { 2 } + 2 y ^ { 2 } = 5$ , then $\frac { d y } { d x } = 2 x$ .

Accepted Answer

To find $\frac{dy}{dx}$, we need to differentiate both sides of the equation $x^2 + 2y^2 = 5$ with respect to $x$, using implicit differentiation. Differentiating $x^2$ with respect to $x$ gives $2x$, and differentiating $2y^2$ with respect to $x$ gives $4y\frac{dy}{dx}$ because of the chain rule. Setting the derivative of the left side equal to the derivative of the right side (which is 0, since the right side is a constant) gives $2x + 4y\frac{dy}{dx} = 0$. Solving for $\frac{dy}{dx}$ does not yield $\frac{dy}{dx} = 2x$, but rather $\frac{dy}{dx} = -\frac{x}{2y}$.

Question 11

Find an equation for the tangent line to the curve $x ^ { 3 } + x y + y ^ { 3 } = x$ at the point (1, 0).

Accepted Answer

The answer of Find an equation for the tangent line...

Question 12

Find an equation for the tangent line to the curve $x ^ { 2 } + y ^ { 3 } = x y + 1$ at the point (1, -1).

Accepted Answer

The answer of Find an equation for the tangent line...

Question 13

Find the equation of the tangent line to the given curve at the specified point: $x ^ { 2 } y ^ { 5 } - 4 x y = 3 x + y - 8$ ; (0, 8)
A) $y = \frac { 1 } { 35 } x + 8$
B) $y = - \frac { 1 } { 35 } x + 8$
C) y = -35x + 8
D) y = 35x + 8

Accepted Answer

The answer of Find the equation of the tangent line...

Question 14

The equation for the tangent line to the curve $x ^ { 2 } + 2 x y = y ^ { 3 }$ at the point (1, -1) is y = -1.

Accepted Answer

The answer of The equation for the tangent line to...

Question 15

Use implicit differentiation to find $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ for $4 x ^ { 5 } + 11 y = 100$ .
A) $80 x ^ { 3 }$
B) $- \frac { 80 } { 11 } x ^ { 3 }$
C) $60 x ^ { 2 } + 11$
D) $60 x ^ { 2 } - 100$

Accepted Answer

To find $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ for $4 x ^ { 5 } + 11 y = 100$, we first differentiate both sides with respect to $x$ to get the first derivative, $y'$. Differentiating $4x^5$ gives $20x^4$, and differentiating $11y$ using the chain rule gives $11y'$. Setting the equation equal to 0 gives $20x^4 + 11y' = 0$, so $y' = -\frac{20}{11}x^4$. Differentiating $y'$ again with respect to $x$ to find $y''$ gives $\frac{d}{dx}(-\frac{20}{11}x^4) = -\frac{80}{11}x^3$, which is option B.

Question 16

In a certain factory, output Q is related to inputs x and y by the equation $Q = 3 x ^ { 3 } + 5 x ^ { 2 } y ^ { 2 } + 2 y ^ { 3 }$ . If the current levels of input are x = 255 and y = 155, use calculus to estimate the change in input y that should be made to offset a decrease of 0.6 unit in input x so that output will be maintained at its current level. Round your answer to two decimal places, if necessary.
A) An increase of 0.37
B) A decrease of 0.37
C) It cannot be determined
D) No change

Accepted Answer

To maintain the output level, we use the concept of total differential: $dQ = \frac{\partial Q}{\partial x} dx + \frac{\partial Q}{\partial y} dy = 0$. Calculate $\frac{\partial Q}{\partial x}$ and $\frac{\partial Q}{\partial y}$, substitute $x = 255$, $y = 155$, and $dx = -0.6$ to solve for $dy$. The result is approximately 0.37, indicating an increase in y.

Question 17

Suppose the output at a certain factory is $Q = 4 x ^ { 2 } + 3 x ^ { 3 } y ^ { 3 } + y ^ { 5 }$ units, where x is the number of hours of skilled labor used and y is the number of hours of unskilled labor. The current labor force consists of 30 hours of skilled labor and 10 hours of unskilled labor. Use calculus to estimate the change in unskilled labor y that should be made to offset a 1-hour increase in skilled labor x so that output will be maintained at its current level. Round you answer to two decimal places, if necessary.
A) -0.33 hours
B) -1 hours
C) -3.01 hours
D) 3.01 hours

Accepted Answer

To maintain the output level when skilled labor x increases by 1 hour, we need to find how unskilled labor y should change. We use the total differential of the output function $Q = 4x^2 + 3x^3y^3 + y^5$, which is $dQ = (8x + 9x^2y^3)dx + (9x^3y^2 + 5y^4)dy$. Setting $dQ = 0$ because the output is to remain constant, and substituting $x = 30$ and $y = 10$, we solve for $dy/dx$ to find the change in y for a unit change in x. The calculation yields $dy/dx \approx -0.33$, indicating that for every 1-hour increase in skilled labor, unskilled labor should decrease by approximately 0.33 hours to maintain the same output level.

Question 18

The equation of the line tangent to the graph of $f ( x ) = x ^ { 2 } + 3 x$ at x = 5 is

Accepted Answer

The answer of The equation of the line tangent to...

Question 19

For f (x) = 6 - x², find the slope of the secant line connecting the points whose x-coordinates are x = -1 and x = -0.9. Then use calculus to find the slope of the line that is tangent to the graph of f at x = -1.

Accepted Answer

The answer of For f (x) = 6 - x²,...

Question 20

If f (x) represents the price per barrel of oil in terms of time, what does $\frac { f \left( x _ { 0 } + h \right) - f \left( x _ { 0 } \right) } { h }$ represent? What about $\lim _ { h \rightarrow 0 } \frac { f \left( x _ { 0 } + h \right) - f \left( x _ { 0 } \right) } { h }$ ?

Accepted Answer

The answer of If f (x) represents the price per...

Quiz 2: Differentiation: Basic Concepts

If the total cost of manufacturing q units of a certain commodity is C(q) = (3q + 1)(5q + 7), use marginal analysis to estimate the cost of producing the 19th unit, in dollars.

If x3+y3=x+yx ^ { 3 } + y ^ { 3 } = x + yx3+y3=x+y , then dydx=3x2−13y2−1\frac { d y } { d x } = \frac { 3 x ^ { 2 } - 1 } { 3 y ^ { 2 } - 1 }dxdy​=3y2−13x2−1​ .

Find dydx\frac { d y } { d x }dxdy​ , where xy3−3x2=7yx y ^ { 3 } - 3 x ^ { 2 } = 7 yxy3−3x2=7y .

Find dydx\frac { d y } { d x }dxdy​ , where x+y=xy\sqrt { x } + \sqrt { y } = x yx​+y​=xy .

Find dydx\frac { d y } { d x }dxdy​ , where 3x+12y=5\frac { 3 } { x } + \frac { 1 } { 2 y } = 5x3​+2y1​=5 .

If x2+3xy+y2=15x ^ { 2 } + 3 x y + y ^ { 2 } = 15x2+3xy+y2=15 , then dydx=2x+3y\frac { d y } { d x } = 2 x + 3 ydxdy​=2x+3y .

If x2y+xy2=7x ^ { 2 } y + x y ^ { 2 } = 7x2y+xy2=7 , then dydx=2xy+y2\frac { d y } { d x } = 2 x y + y ^ { 2 }dxdy​=2xy+y2 .

If x2+2y2=5x ^ { 2 } + 2 y ^ { 2 } = 5x2+2y2=5 , then dydx=2x\frac { d y } { d x } = 2 xdxdy​=2x .

Find an equation for the tangent line to the curve x3+xy+y3=xx ^ { 3 } + x y + y ^ { 3 } = xx3+xy+y3=x at the point (1, 0).

Find an equation for the tangent line to the curve x2+y3=xy+1x ^ { 2 } + y ^ { 3 } = x y + 1x2+y3=xy+1 at the point (1, -1).

Find the equation of the tangent line to the given curve at the specified point: x2y5−4xy=3x+y−8x ^ { 2 } y ^ { 5 } - 4 x y = 3 x + y - 8x2y5−4xy=3x+y−8 ; (0, 8)

The equation for the tangent line to the curve x2+2xy=y3x ^ { 2 } + 2 x y = y ^ { 3 }x2+2xy=y3 at the point (1, -1) is y = -1.

Use implicit differentiation to find d2ydx2\frac { d ^ { 2 } y } { d x ^ { 2 } }dx2d2y​ for 4x5+11y=1004 x ^ { 5 } + 11 y = 1004x5+11y=100 .

The equation of the line tangent to the graph of f(x)=x2+3xf ( x ) = x ^ { 2 } + 3 xf(x)=x2+3x at x = 5 is

For f (x) = 6 - x2, find the slope of the secant line connecting the points whose x-coordinates are x = -1 and x = -0.9. Then use calculus to find the slope of the line that is tangent to the graph of f at x = -1.