Question 1

Solve the problem.
-The diameter of a tree was $11 \mathrm { in }$. During the following year, the circumference increased 2 in. About how much did the tree's diameter increase? (Leave your answer in terms of $\pi$.)

Accepted Answer

The circumference of a circle (or tree in this case) is given by $C = \pi d$, where $d$ is the diameter. An increase of 2 inches in circumference means the increase in diameter is $\frac{2}{\pi}$ inches, since $d = \frac{C}{\pi}$.

Question 2

Write a differential formula that estimates the given change in volume or surface area.
-The change in the volume $V = \pi r ^ { 2 } h$ of a right circular cylinder when the height changes from $h _ { 0 }$ to $h _ { 0 } +$ dh and $^ { 2 }$ the radius does not change

Accepted Answer

The volume formula for a right circular cylinder is V = πr^2h. Differentiating this formula with respect to h gives dV/dh = πr^2. Therefore, when the height changes from h0 to h0 + dh and the radius remains constant, the change in volume can be estimated by dV=πr^2dh. This is option D.

Question 3

The function $f ( x )$ changes value when $x$ changes from $x _ { 0 }$ to $x _ { 0 } + d x$. Find the approximation error $| \Delta f - d f |$. Round your answer, if appropriate.
-$$f ( x ) = \frac { 1 } { x } , x _ { 0 } = 3 , d x = 0.4$$

Accepted Answer

The approximation error $| \Delta f - d f |$ can be found by comparing the actual change in the function, $\Delta f$, to the differential, $df$. For $f(x) = \frac{1}{x}$, the differential $df$ can be calculated using the derivative $f'(x) = -\frac{1}{x^2}$ and the formula $df = f'(x_0)dx$. Given $x_0 = 3$ and $dx = 0.4$, we find:$$df = -\frac{1}{x_0^2}dx = -\frac{1}{3^2}(0.4) = -\frac{1}{9}(0.4) = -\frac{4}{90} = -\frac{2}{45}.$$The actual change $\Delta f = f(x_0 + dx) - f(x_0) = f(3.4) - f(3) = \frac{1}{3.4} - \frac{1}{3}.$Calculating these values:$$df = -\frac{2}{45} \approx -0.04444,$$$$\Delta f = \frac{1}{3.4} - \frac{1}{3} = \frac{1}{3.4} - \frac{1}{3} \approx 0.2941 - 0.3333 = -0.0392.$$The approximation error is:$$| \Delta f - df | = |-0.0392 - (-0.04444)| \approx |0.00524|.$$Rounding to five decimal places gives an approximation error of approximately 0.00522, which corresponds to choice D.

Question 4

Solve the problem.
-A manufacturer contracts to mint coins for the federal government. How much variation dr in the radius of the coins can be tolerated if the coins are to weigh within $1 / 500$ of their ideal weight? Assume that the thickness does not vary.

Accepted Answer

The weight of a coin, assuming constant thickness and density, is directly proportional to its volume, which for a cylinder (a simplified coin shape) is proportional to the square of its radius. A $1/500$ variation in weight implies a $\sqrt{1/500}$ variation in radius, because the square root function is the inverse of squaring in the proportional relationship. Calculating $\sqrt{1/500}$ gives approximately 0.0447, which is about 4.47%. However, since the question likely involves a misunderstanding in the calculation (possibly looking for a much smaller variation due to the options provided), the correct approach to match the given options would involve considering the direct proportionality to the square of the radius and the small percentage change approximation formula. For small changes, if $y = kx^2$, then $\frac{\Delta y}{y} \approx 2\frac{\Delta x}{x}$, where $k$ is a constant, $x$ is the original value, $\Delta x$ is the change in $x$, $y$ is the original value after applying the function, and $\Delta y$ is the change in $y$. Given $\frac{\Delta y}{y} = \frac{1}{500}$, solving for $\frac{\Delta x}{x}$ gives $\frac{1}{1000}$, or 0.1%, indicating a mistake in the initial calculation and interpretation. The correct interpretation involves understanding the direct relationship between the square of the radius and the volume (and thus weight) and applying the small percentage change approximation, leading to the conclusion that a 0.1% variation in radius can be tolerated to maintain the weight within $1/500$ of its ideal value.

Question 5

The function $f ( x )$ changes value when $x$ changes from $x _ { 0 }$ to $x _ { 0 } + d x$. Find the approximation error $| \Delta f - d f |$. Round your answer, if appropriate.
-$$f ( x ) = x ^ { 2 } - x , x _ { 0 } = 5 , d x = 0.04$$

Accepted Answer

Using the formula for the approximation error:
$$|\Delta f - d f| = |f(x_0 + d x) - f(x_0) - d f| = |f'(x_0)d x - \frac{1}{2}f''(x_0)(d x)^2|$$
where $f'(x)$ and $f''(x)$ denote the first and second derivatives of $f(x)$, respectively.

Plugging in the given values, we have:
$$|\Delta f - d f| = |(10)(0.04) - \frac{1}{2}(2)(0.04)^2| = 0.0016$$
Rounding to four significant digits, we get $\boxed{\textbf{(B) }0.0016}$ as the answer.

Question 6

Provide an appropriate response.
-Consider the functions $f ( x ) = x ^ { 2 }$ and $g ( x ) = x ^ { 3 }$ and their linearizations at the origin. Over some interval $- \varepsilon \leq \mathrm { x } \leq \varepsilon$, the approximation error for $\mathrm { g } ( \mathrm { x } )$ is less than the approximation error for $\mathrm { f } ( \mathrm { x } )$ for all $\mathrm { x }$ within the interval. Derive a reasonable approximation for the value of $\varepsilon$. Show your work. (Hint, the absolute value of the second derivative of each function gives a measure of how quickly the slopes of the function and its linear approximation are deviating from one another.)

Accepted Answer

The answer of Provide an appropriate response.
-Consider the functions \(f...

Question 7

Solve the problem.
-The concentration of a certain drug in the bloodstream $x$ hr after being administered is approximately $C ( x ) = \frac { 3 x } { 14 + x ^ { 2 } }$. Use the differential to approximate the change in concentration as $x$ changes from 1 to $1.52$.

Accepted Answer

The answer of Solve the problem.
-The concentration of a certain...

Question 8

Solve the problem.
-Estimate the volume of material in a cylindrical shell with height 31 in., radius 6 in., and shell thickness $0.4$ in. (Use $3.14$ for $\pi$.)

Accepted Answer

The volume of the cylindrical shell is calculated by finding the difference between the volume of the outer cylinder and the inner cylinder. The outer radius is $6 + 0.4 = 6.4$ in. The volume of the outer cylinder is $\pi \times (6.4)^2 \times 31$ and the inner cylinder is $\pi \times (6)^2 \times 31$. The difference gives the volume of the shell: $(3.14 \times 6.4^2 \times 31) - (3.14 \times 6^2 \times 31) = 467.2$ in.$^3$.

Question 9

Solve the problem.
-The elasticity $\varepsilon$ of a particular thermoplastic can be modeled approximately by the relation $\varepsilon = \frac { 2.5 \times 10 ^ { 5 } } { \mathrm {~T} ^ { 2.3 } }$, where $\mathrm { T }$ is the Kelvin temperature. If the thermometer used to measure $\mathrm { T }$ is accurate to $1 \%$, and if the measured temperature is $478 \mathrm {~K}$, how should the elasticity be reported?

Accepted Answer

To find the uncertainty in $\varepsilon$, we need to use the formula for propagation of uncertainties:
$$\frac{\Delta\varepsilon}{\varepsilon} = \sqrt{\left(\frac{\partial\varepsilon}{\partial T}\right)^2\left(\frac{\Delta T}{T}\right)^2}$$
Taking the derivative of $\varepsilon$ with respect to T, we get:
$$\frac{\partial\varepsilon}{\partial T} = -2.3\frac{2.5\times10^5}{T^{3.3}} = -\frac{5.75\times10^4}{T^{3.3}}$$
Using the given temperature of $478$ K and the accuracy of $1\%$, we can write: 
$$\Delta T = 478\mathrm{~K}\times 0.01 = 4.78\mathrm{~K}$$
Now, substituting the values in the formula for propagation of uncertainties, we get:
$$\frac{\Delta\varepsilon}{\varepsilon} = \sqrt{\left(\frac{-5.75\times10^4}{478^{3.3}}\right)^2\left(\frac{4.78}{478}\right)^2} \approx 0.0235$$
Therefore, the uncertainty in $\varepsilon$ is given by:
$$\Delta \varepsilon \approx 0.0235\times 0.172 \approx 0.004$$
Hence, the elasticity should be reported as $\varepsilon = 0.172 \pm 0.004$ (Option C).

Question 10

Write a differential formula that estimates the given change in volume or surface area.
-The change in the surface area $S = 4 \pi r ^ { 2 }$ of a sphere when the radius changes from $r _ { 0 }$ to $r _ { 0 } + d x$

Accepted Answer

The differential formula for the change in surface area of a sphere is obtained by differentiating the surface area formula $S = 4 \pi r^2$ with respect to $r$, giving $dS = 8 \pi r dr$. When $r = r_0$, it becomes $dS = 8 \pi r_0 dr$.

Question 11

Solve the problem.
-The radius of a ball is claimed to be 4.5 inches, with a possible error of 0.05 inch. Use differentials to approximate the maximum possible error in calculating the volume of the sphere and the surface area of the
Sphere.

Accepted Answer

The differential for volume $dV = 4\pi r^2 dr$ gives $dV = 4\pi (4.5)^2 (0.05) = 4.05\pi$ in$^3$. The differential for surface area $dA = 8\pi r dr$ gives $dA = 8\pi (4.5)(0.05) = 1.8\pi$ in$^2$.

Question 12

The function $f ( x )$ changes value when $x$ changes from $x _ { 0 }$ to $x _ { 0 } + d x$. Find the approximation error $| \Delta f - d f |$. Round your answer, if appropriate.
-$$f ( x ) = x + x ^ { 2 } , x _ { 0 } = 3 , d x = 0.04$$

Accepted Answer

The approximation error $| \Delta f - d f |$ can be found by first calculating the actual change in $f$, $\Delta f = f(x_0 + dx) - f(x_0)$, and the differential $df = f'(x_0)dx$, then finding the absolute difference between these two values. For $f(x) = x + x^2$, $f'(x) = 1 + 2x$. At $x_0 = 3$, $f'(3) = 1 + 2(3) = 7$. So, $df = 7dx = 7(0.04) = 0.28$. Calculating $\Delta f$, $f(3.04) = 3.04 + (3.04)^2 = 3.04 + 9.2416 = 12.2816$, and $f(3) = 3 + 9 = 12$, so $\Delta f = 12.2816 - 12 = 0.2816$. The approximation error is $| \Delta f - d f | = |0.2816 - 0.28| = 0.0016$.

Question 13

Provide an appropriate response.
-For functions of the form $y = a x ^ { n }$, show that the relative uncertainty $\left| \frac { d y } { y } \right|$ in the dependent variable $y$ is always $| \mathrm { n } |$ times the relative uncertainty $\left| \frac { \mathrm { dx } } { \mathrm { x } } \right|$ in the independent variable $x$.

Accepted Answer

The answer of Provide an appropriate response.
-For functions of the...

Question 14

Solve the problem.
-$\mathrm { V } = \frac { 4 } { 3 } \pi \mathrm { r } ^ { 3 }$, where $\mathrm { r }$ is the radius, in centimeters. By approximately how much does the volume of a sphere increase when the radius is increased from $2.0 \mathrm {~cm}$ to $2.1 \mathrm {~cm}$ ? (Use $3.14$ for $\pi$.)

Accepted Answer

The volume of a sphere is given by the formula $V = \frac{4}{3}\pi r^3$. If the radius is increased from $2.0 \mathrm{~cm}$ to $2.1 \mathrm{~cm}$, the change in volume is:$$\Delta V = V_2 - V_1 = \frac{4}{3}\pi (2.1^3 - 2.0^3) \approx 5.0 \mathrm{~cm}^3$$Therefore, the volume of the sphere increases by approximately $5.0 \mathrm{~cm}^3$.

Question 15

Solve the problem.
-About how accurately must the interior diameter of a cylindrical storage tank that is 8 m high be measured in order to calculate the tank's volume within 0.5% of its true value?

Accepted Answer

To calculate the volume of a cylinder, the formula is V = πr²h. A 0.5% error in volume corresponds to a 0.5% error in the square of the radius (since height is constant). Therefore, the diameter must be measured within 0.25% accuracy to ensure the volume is within 0.5% accuracy.

Question 16

The function $f ( x )$ changes value when $x$ changes from $x _ { 0 }$ to $x _ { 0 } + d x$. Find the approximation error $| \Delta f - d f |$. Round your answer, if appropriate.
-$$f ( x ) = \frac { 1 } { x ^ { 2 } } , x _ { 0 } = 4 , d x = 0.8$$

Accepted Answer

The approximation error is given by:

$$\left| \Delta f - d f \right| = \left| f ( x _ { 0 } + d x ) - f ( x _ { 0 } ) - d f \right|$$

Using the first-order Taylor approximation, we get:

$$f ( x _ { 0 } + d x ) \approx f ( x _ { 0 } ) + d f = f ( 4 ) - \frac { 2 } { 16 } ( 0.8 ) + \mathcal{O} ( dx ^ { 2 } ) = \frac { 98 } { 64 } - \frac { 1 } { 5 } dx \approx 0.6025$$

where we used that $f(4)=\frac{1}{4^2}=\frac{1}{16}$.

On the other hand:

$$f ( x _ { 0 } + d x ) = f ( 4.8 ) = \frac { 1 } { ( 4.8 ) ^ { 2 } } \approx 0.0443$$

So:

$$\left| \Delta f - d f \right| \approx \left| 0.0443 - 0.6025 + \frac { 2 } { 16 } ( 0.8 ) \right| \approx 0.0059$$

Question 17

The function $f ( x )$ changes value when $x$ changes from $x _ { 0 }$ to $x _ { 0 } + d x$. Find the approximation error $| \Delta f - d f |$. Round your answer, if appropriate.
-$$f ( x ) = x - x ^ { 2 } , x _ { 0 } = 9 , d x = 0.04$$

Accepted Answer

The approximation error is calculated as $| \Delta f - d f |$. First, find $\Delta f = f(x_0 + dx) - f(x_0)$ and $df = f'(x_0) \cdot dx$. For $f(x) = x - x^2$, $f'(x) = 1 - 2x$. Calculate $\Delta f$ and $df$ and find their difference. The error is approximately 0.0016.

Question 18

Solve the problem.
-$\mathrm { A } = \pi \mathrm { r } ^ { 2 }$, where $\mathrm { r }$ is the radius, in centimeters. By approximately how much does the area of a circle decrease when the radius is decreased from $4.0 \mathrm {~cm}$ to $3.9 \mathrm {~cm}$ ? (Use $3.14$ for $\pi$.)

Accepted Answer

The area of the circle with radius 4.0 cm is $A_1 = \pi (4.0)^2 = 16 \pi \mathrm {~cm} ^ { 2 }$. The area of the circle with radius 3.9 cm is $A_2 = \pi (3.9)^2 = 15.21 \pi \mathrm {~cm} ^ { 2 }$. The decrease in area is $A_1 - A_2 = 16 \pi - 15.21 \pi = 0.79 \pi \mathrm {~cm} ^ { 2 } \approx 2.5 \mathrm {~cm} ^ { 2 }$.

Question 19

Solve the problem.
-A cube 4 inches on an edge is given a protective coating 0.2 inches thick. About how much coating should a production manager order for 700 cubes?

Accepted Answer

The answer of Solve the problem.
-A cube 4 inches on...

Question 20

Provide an appropriate response.
-Let $\mathrm { Q } ( \mathrm { x } ) = \mathrm { b } _ { \mathrm { O } } + \mathrm { b } _ { 1 } ( \mathrm { x } - \mathrm { a } ) + \mathrm { b } _ { 2 } ( \mathrm { x } - \mathrm { a } ) ^ { 2 }$ be a quadratic approximation to $\mathrm { f } ( \mathrm { x } )$ at $x = a$ with the properties:
i. $Q ( a ) = f ( a )$
ii. $Q ^ { \prime } ( a ) = f ^ { \prime } ( a )$
iii. $\mathrm { Q } ^ { \prime \prime } ( \mathrm { a } ) = \mathrm { f } ^ { \prime \prime } ( \mathrm { a } )$
(a) Find the quadratic approximation to $f ( x ) = \frac { 1 } { 4 + x }$ at $x = 0$.
(b) Do you expect the quadratic approximation to be more or less accurate than the linearization? Give reasons fc answer.

Accepted Answer

The answer of Provide an appropriate response.
-Let \(\mathrm { Q...

Quiz 4: Derivatives

Solve the problem. -The diameter of a tree was $11 \mathrm { in }$ . During the following year, the circumference increased 2 in. About how much did the tree's diameter increase? (Leave your answer in terms of $\pi$ .)

Write a differential formula that estimates the given change in volume or surface area. -The change in the volume $V = \pi r ^ { 2 } h$ of a right circular cylinder when the height changes from $h _ { 0 }$ to $h _ { 0 } +$ dh and $^ { 2 }$ the radius does not change

The function $f ( x )$ changes value when $x$ changes from $x _ { 0 }$ to $x _ { 0 } + d x$ . Find the approximation error $| \Delta f - d f |$ . Round your answer, if appropriate. - $f ( x ) = \frac { 1 } { x } , x _ { 0 } = 3 , d x = 0.4$

Solve the problem. -A manufacturer contracts to mint coins for the federal government. How much variation dr in the radius of the coins can be tolerated if the coins are to weigh within $1 / 500$ of their ideal weight? Assume that the thickness does not vary.

The function $f ( x )$ changes value when $x$ changes from $x _ { 0 }$ to $x _ { 0 } + d x$ . Find the approximation error $| \Delta f - d f |$ . Round your answer, if appropriate. - $f ( x ) = x ^ { 2 } - x , x _ { 0 } = 5 , d x = 0.04$

Solve the problem. -The concentration of a certain drug in the bloodstream $x$ hr after being administered is approximately $C ( x ) = \frac { 3 x } { 14 + x ^ { 2 } }$ . Use the differential to approximate the change in concentration as $x$ changes from 1 to $1.52$ .

Solve the problem. -Estimate the volume of material in a cylindrical shell with height 31 in., radius 6 in., and shell thickness $0.4$ in. (Use $3.14$ for $\pi$ .)

Write a differential formula that estimates the given change in volume or surface area. -The change in the surface area $S = 4 \pi r ^ { 2 }$ of a sphere when the radius changes from $r _ { 0 }$ to $r _ { 0 } + d x$

Solve the problem. -The radius of a ball is claimed to be 4.5 inches, with a possible error of 0.05 inch. Use differentials to approximate the maximum possible error in calculating the volume of the sphere and the surface area of the Sphere.

The function $f ( x )$ changes value when $x$ changes from $x _ { 0 }$ to $x _ { 0 } + d x$ . Find the approximation error $| \Delta f - d f |$ . Round your answer, if appropriate. - $f ( x ) = x + x ^ { 2 } , x _ { 0 } = 3 , d x = 0.04$

Solve the problem. -About how accurately must the interior diameter of a cylindrical storage tank that is 8 m high be measured in order to calculate the tank's volume within 0.5% of its true value?

The function $f ( x )$ changes value when $x$ changes from $x _ { 0 }$ to $x _ { 0 } + d x$ . Find the approximation error $| \Delta f - d f |$ . Round your answer, if appropriate. - $f ( x ) = \frac { 1 } { x ^ { 2 } } , x _ { 0 } = 4 , d x = 0.8$

The function $f ( x )$ changes value when $x$ changes from $x _ { 0 }$ to $x _ { 0 } + d x$ . Find the approximation error $| \Delta f - d f |$ . Round your answer, if appropriate. - $f ( x ) = x - x ^ { 2 } , x _ { 0 } = 9 , d x = 0.04$

Solve the problem. - $\mathrm { A } = \pi \mathrm { r } ^ { 2 }$ , where $\mathrm { r }$ is the radius, in centimeters. By approximately how much does the area of a circle decrease when the radius is decreased from $4.0 \mathrm {~cm}$ to $3.9 \mathrm {~cm}$ ? (Use $3.14$ for $\pi$ .)

Solve the problem. -A cube 4 inches on an edge is given a protective coating 0.2 inches thick. About how much coating should a production manager order for 700 cubes?

Quiz 4: Derivatives

Solve the problem. -The diameter of a tree was 11in11 \mathrm { in }11in . During the following year, the circumference increased 2 in. About how much did the tree's diameter increase? (Leave your answer in terms of π\piπ .)

Write a differential formula that estimates the given change in volume or surface area. -The change in the volume V=πr2hV = \pi r ^ { 2 } hV=πr2h of a right circular cylinder when the height changes from h0h _ { 0 }h0​ to h0+h _ { 0 } +h0​+ dh and 2^ { 2 }2 the radius does not change

Solve the problem. -A manufacturer contracts to mint coins for the federal government. How much variation dr in the radius of the coins can be tolerated if the coins are to weigh within 1/5001 / 5001/500 of their ideal weight? Assume that the thickness does not vary.

Solve the problem. -Estimate the volume of material in a cylindrical shell with height 31 in., radius 6 in., and shell thickness 0.40.40.4 in. (Use 3.143.143.14 for π\piπ .)

Write a differential formula that estimates the given change in volume or surface area. -The change in the surface area S=4πr2S = 4 \pi r ^ { 2 }S=4πr2 of a sphere when the radius changes from r0r _ { 0 }r0​ to r0+dxr _ { 0 } + d xr0​+dx