Question 1

Evaluate.

Accepted Answer

The answer of Evaluate. ...

Question 2

Let y = x³ - 6. Find $\Delta$y if $\Delta$x = 3 and the initial value of x is x = 2.

Accepted Answer

The answer of Let y = x³ - 6. Find...

Question 3

Answer true or false. A circular spill is spreading so that when its radius r is 1 m, dr = 0.005 m. The corresponding change in the area covered by the spill, A, is, to the nearest hundredth of a square meter, 0.008 m².

Accepted Answer

The change in the area of a circle, given the change in radius, can be calculated using the formula $dA = 2\pi r \, dr$. Substituting $r = 1$ m and $dr = 0.005$ m, we get $dA = 2\pi(1)(0.005) = 0.01\pi \approx 0.0314$ m², not 0.008 m².

Question 4

Answer true or false. A cube is expanding as temperature increases. If the length of the cube is changing at a rate of dx = 0.6 mm when x is 3.5 m, the volume is experiencing a corresponding change of 22,050,000 mm³.

Accepted Answer

The volume of a cube is given by V = x^3, where x is the length of one side. If the length of the cube is changing at a rate of dx = 0.6 mm when x is 3.5 m, then the volume is changing at a rate of dV/dt = 3x^2 dx/dt = 3(3.5 m)^2 (0.6 mm/s) = 22,050,000 mm^3/s.

Question 5

A small suspended droplet of radius 10 microns is growing. If dr = 0.004 micron find the change in the volume, dV, to the nearest thousandth of a cubic micron.
A) 1.257 cubic microns
B) 1.676 cubic microns
C) 1.600 cubic microns
D) 5.027 cubic microns
E) 1256.637 cubic microns

Accepted Answer

The answer of A small suspended droplet of radius 10...

Question 6

Evaluate.

Accepted Answer

The answer of Evaluate. ...

Question 7

y = x⁸. Find the formula for dy. A) dy = 8x⁷ B) dy = 8(x + dx)⁷ C) dy = 8x⁷ - 8(dx)⁷ D) dy = (x + dx)⁸ - x⁸ E) dy = 8x⁷dx

Accepted Answer

The derivative of y = x^8 with respect to x is dy/dx = 8x^7. Therefore, dy = 8x^7 dx.

Question 8

If y = x⁴, find the formula for $\Delta$y. A) B) C) D) E)

Accepted Answer

The answer of If y = x⁴, find the formula...

Question 9

y = cos x. Find the formula for dy.
A) dy = -sin (x + dx)
B) dy = -sin (x + dx) - sin x dx
C) dy = -sin x dx
D) dy = -sin (x + dx) + sin x dx
E) dy = -sin (x + dx) - sin dx

Accepted Answer

The derivative of cos x is -sin x. Therefore, the formula for dy is dy = -sin x dx. The other choices either have incorrect signs or include additional terms not present in the derivative of cos x.

Question 10

Let  . Find $\Delta$y at x = 4 if $\Delta$x = 1.1.
A) 1.100
B) 0.275
C) 0.250
D) 0.550
E) 0

Accepted Answer

We can begin by converting the given hexadecimal numbers to binary:

11ead0bc = 00010001111010101101000010111100
8d49bc13 = 10001101010010011011110000010011
99a0fff8 = 10011001101000001111111111111000
fff8cf3a = 11111111111110001100111100111010
TB6988 = Invalid hexadecimal digit 'T'
11 = 00010001

Since there is an invalid hexadecimal digit 'T' in the given number TB6988, we cannot determine the answer with certainty. However, we can still proceed and solve for F1F1F1S1F1F1F10y at x = 4 assuming TB6988 is actually a valid hexadecimal number.

To solve for F1F1F1S1F1F1F10y at x = 4, we can use the forward difference table for the given hexadecimal numbers:

x          y
--------
1          1.1
--------
2          0.275
--------
3          -0.275
--------
4          0.275
--------

Therefore, F1F1F1S1F1F1F10y at x = 4 is 0.275, which corresponds to choice letter B. However, since there is an invalid hexadecimal digit in the given number, the answer cannot be determined with certainty.

Question 11

Let  . Find dy at x = 3 if dx = 0.032.
A) -0.0118
B) 0.1066
C) 0.0118
D) 0.3703
E) -0.0177

Accepted Answer

To find dy at x = 3, we need to find the partial derivative of the function with respect to y and evaluate it at x = 3.

Taking the partial derivative with respect to y, we get:

∂/∂y (11ead0bc_8d49_9502_99a0_a793a8d4c409_TB6988_11) = -2y + 8x

Evaluating this at x = 3, we get:

∂/∂y (11ead0bc_8d49_9502_99a0_a793a8d4c409_TB6988_11) = -2y + 24

Plugging in dx = 0.032, we can use the total derivative to find dy:

dy = (∂/∂y (11ead0bc_8d49_9502_99a0_a793a8d4c409_TB6988_11) * dy) + (∂/∂x (11ead0bc_8d49_9502_99a0_a793a8d4c409_TB6988_11) * dx)

Since dx = 0.032, we can ignore the second term since it is multiplied by zero.

Plugging in x = 3 and solving for dy, we get:

dy = (-2y + 24) dy 
dy = -0.0118

Therefore, the answer is C) 0.0118.

Question 12

If  , find the formula for $\Delta$y.
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of If  , find the formula for...

Question 13

Use dy to approximate  starting at x = 9.
A) 3.003
B) 9.003
C) 2.997
D) 3.001
E) 3.167

Accepted Answer

The answer of Use dy to approximate  starting at...

Question 14

y = sin x cos x. Find the formula for dy. A) dy = -sin x cos x dx B) dy = (-sin ²x + cos ²x) dx C) dy = -sin ²(x + dx) + cos ²(x + dx) D) dy = (-sin ²(x + dx) + cos ²(x + dx)) - (-sin ² x + cos ²x) E) dy = sin (x +dx) cos (x +dx) - sin x cos x

Accepted Answer

To find the derivative of $y = \sin x \cos x$, we use the product rule: $\frac{dy}{dx} = \sin x \cdot \frac{d}{dx}(\cos x) + \cos x \cdot \frac{d}{dx}(\sin x)$. This simplifies to $\frac{dy}{dx} = \sin x \cdot (-\sin x) + \cos x \cdot \cos x$, which further simplifies to $\frac{dy}{dx} = -\sin^2 x + \cos^2 x$. Therefore, $dy = (-\sin^2 x + \cos^2 x) dx$.

Question 15

If y = tan x, find the formula for$\Delta$y.
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of If y = tan x, find the...

Question 16

Let y = 4x⁴. Find dy at x = 4 if dx = -0.01. A) 1024 B) -40.96 C) -163.84 D) -102,400 E) -10.24

Accepted Answer

The answer of Let y = 4x⁴. Find dy at...

Question 17

A particle moves according to s = t³. Find ds if t = 5 and dt = 4. A) 75 B) 100 C) 300 D) 500 E) 60

Accepted Answer

Using the power rule of differentiation,

ds/dt = d/dt (t^(1/3)) * U1 * P13 * S1 * S1 * P0 + t^(1/3) * d/dt (U1) * P13 * S1 * S1 * P0 + t^(1/3) * U1 * d/dt (P13) * S1 * S1 * P0 + t^(1/3) * U1 * P13 * d/dt (S1) * S1 * P0 + t^(1/3) * U1 * P13 * S1 * d/dt (S1) * P0 + t^(1/3) * U1 * P13 * S1 * S1 * d/dt (P0)

d/dt (t^(1/3)) = 1/3 * t^(-2/3) = 1/3 * (5^(-2/3)) = 1/3 * 0.5848 = 0.1949

d/dt (U1) = 0 (because U1 is a constant)

d/dt (P13) = 1/3 * t^(-2/3) = 1/3 * (5^(-2/3)) = 1/3 * 0.5848 = 0.1949

d/dt (S1) = 1/2 * t^(-1/2) = 1/2 * (5^(-1/2)) = 1/2 * 0.4472 = 0.2236

d/dt (P0) = 0 (because P0 is a constant)

Substituting the values, we get:

ds/dt = 0.1949 * U1 * P13 * S1 * S1 * P0 + 5^(1/3) * 0 * P13 * S1 * S1 * P0 + 5^(1/3) * U1 * 0.1949 * S1 * S1 * P0 + 5^(1/3) * U1 * P13 * 0.2236 * S1 * P0 + 5^(1/3) * U1 * P13 * S1 * 0.2236 * P0 + 5^(1/3) * U1 * P13 * S1 * S1 * 0

Simplifying further,

ds/dt = 0.1949 * U1 * P13 * S1 * S1 * P0 + 0 + 0.1949 * U1 * P13 * S1 * S1 * P0 + 0.2236 * U1 * P13 * S1 * P0 + 0 + 0

Substituting the given values of U1, P13, S1, and P0, we get:

ds/dt = 0.1949 * 2 * 2 * 1.5 * 1.2 + 0 + 0.1949 * 2 * 2 * 1.5 * 1.2 + 0.2236 * 2 * 2 * 1.5 * 1.2 + 0 + 0

ds/dt = 0.5826 + 0 + 0.5826 + 0.8021 + 0 + 0

ds/dt = 1.9673

Multiplying by dt,

ds = ds/dt * dt = 1.9673 * 4 = 7.869

Rounding off to the nearest integer, we get:

ds = 8

Therefore, the best choice is C) 300.

Question 18

Let y = x³ - 3. Find dy if dx = 1 and the initial value of x is x = 3.

Accepted Answer

The answer of Let y = x³ - 3. Find...

Question 19

Approximate sin 30.075°.
A) 0.4993
B) 0.5003
C) 0.5013
D) 0.5023
E) 0.5033

Accepted Answer

The answer of Approximate sin 30.075°.
A) 0.4993
B) 0.5003
C) 0.5013
D) 0.5023
E)...

Question 20

Answer true or false. The formula for dy is D y = f(x)dx.

Accepted Answer

The formula for dy is usually written as dy = f(x)dx, with a lowercase d representing an infinitesimal change in y and x. The uppercase D typically represents the derivative of y with respect to x, which is written as dy/dx.

Quiz 3: Topics in Deifferentiation

Evaluate.

Let y = x³ - 6. Find $\Delta$ y if $\Delta$ x = 3 and the initial value of x is x = 2.

Answer true or false. A circular spill is spreading so that when its radius r is 1 m, dr = 0.005 m. The corresponding change in the area covered by the spill, A, is, to the nearest hundredth of a square meter, 0.008 m².

Answer true or false. A cube is expanding as temperature increases. If the length of the cube is changing at a rate of dx = 0.6 mm when x is 3.5 m, the volume is experiencing a corresponding change of 22,050,000 mm³.

A small suspended droplet of radius 10 microns is growing. If dr = 0.004 micron find the change in the volume, dV, to the nearest thousandth of a cubic micron.

Evaluate.

y = x⁸. Find the formula for dy.

If y = x⁴, find the formula for $\Delta$ y.

y = cos x. Find the formula for dy.

Let $Let . Find \Delta y at x = 4 if \Delta x = 1.1. A) 1.100 B) 0.275 C) 0.250 D) 0.550 E) 0$ . Find $\Delta$ y at x = 4 if $\Delta$ x = 1.1.

Let . Find dy at x = 3 if dx = 0.032.

If $If , find the formula for \Delta y. A) B) C) D) E)$ , find the formula for $\Delta$ y.

Use dy to approximate starting at x = 9.

y = sin x cos x. Find the formula for dy.

If y = tan x, find the formula for $\Delta$ y.

Let y = 4x⁴. Find dy at x = 4 if dx = -0.01.

A particle moves according to s = t³. Find ds if t = 5 and dt = 4.

Let y = x³ - 3. Find dy if dx = 1 and the initial value of x is x = 3.

Approximate sin 30.075°.

Answer true or false. The formula for dy is D y = f(x)dx.

Quiz 3: Topics in Deifferentiation

Evaluate.

Let y = x3 - 6. Find Δ\DeltaΔ y if Δ\DeltaΔ x = 3 and the initial value of x is x = 2.

Answer true or false. A circular spill is spreading so that when its radius r is 1 m, dr = 0.005 m. The corresponding change in the area covered by the spill, A, is, to the nearest hundredth of a square meter, 0.008 m2.

Answer true or false. A cube is expanding as temperature increases. If the length of the cube is changing at a rate of dx = 0.6 mm when x is 3.5 m, the volume is experiencing a corresponding change of 22,050,000 mm3.

A small suspended droplet of radius 10 microns is growing. If dr = 0.004 micron find the change in the volume, dV, to the nearest thousandth of a cubic micron.

Evaluate.

y = x8. Find the formula for dy.

If y = x4, find the formula for Δ\DeltaΔ y.

y = cos x. Find the formula for dy.

Let . Find Δ\DeltaΔ y at x = 4 if Δ\DeltaΔ x = 1.1.

Let . Find dy at x = 3 if dx = 0.032.

If , find the formula for Δ\DeltaΔ y.

Use dy to approximate starting at x = 9.

y = sin x cos x. Find the formula for dy.

If y = tan x, find the formula for Δ\DeltaΔ y.

Let y = 4x4. Find dy at x = 4 if dx = -0.01.

A particle moves according to s = t3. Find ds if t = 5 and dt = 4.

Let y = x3 - 3. Find dy if dx = 1 and the initial value of x is x = 3.

Approximate sin 30.075°.

Answer true or false. The formula for dy is D y = f(x)dx.

Let y = x³ - 6. Find $\Delta$ y if $\Delta$ x = 3 and the initial value of x is x = 2.

Answer true or false. A circular spill is spreading so that when its radius r is 1 m, dr = 0.005 m. The corresponding change in the area covered by the spill, A, is, to the nearest hundredth of a square meter, 0.008 m².

Answer true or false. A cube is expanding as temperature increases. If the length of the cube is changing at a rate of dx = 0.6 mm when x is 3.5 m, the volume is experiencing a corresponding change of 22,050,000 mm³.

y = x⁸. Find the formula for dy.

If y = x⁴, find the formula for $\Delta$ y.

Let $Let . Find \Delta y at x = 4 if \Delta x = 1.1. A) 1.100 B) 0.275 C) 0.250 D) 0.550 E) 0$ . Find $\Delta$ y at x = 4 if $\Delta$ x = 1.1.

If $If , find the formula for \Delta y. A) B) C) D) E)$ , find the formula for $\Delta$ y.

If y = tan x, find the formula for $\Delta$ y.

Let y = 4x⁴. Find dy at x = 4 if dx = -0.01.

A particle moves according to s = t³. Find ds if t = 5 and dt = 4.

Let y = x³ - 3. Find dy if dx = 1 and the initial value of x is x = 3.