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In a right-angled triangle, the lengths of the two legs are a = 15 and b = 9. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 15^2 + 9^2 = c^2
2. Calculate the squares: 225 + 81 = c^2
3. Sum the squares: 306 = c^2
4. Take the square root of both sides: c = √306
Final Ans... |
Find the derivative of f(x) = (5x + 4)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 5x + 4.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 5x + 4. Its derivative is h'(x) = 5.
4. Multiply g'(u) by h'(x): f'(x) = 3(5x + 4)^2 * 5 = 15(5x ... |
Find the derivative of f(x) = (5x + 4)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 5x + 4.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 5x + 4. Its derivative is h'(x) = 5.
4. Multiply g'(u) by h'(x): f'(x) = 3(5x + 4)^2 * 5 = 15(5x ... |
Add the fractions 13/9 and 8/4. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 9 and 4, which is 36.
2. Convert 13/9 by multiplying top and bottom by 4: 52/36.
3. Convert 8/4 by multiplying top and bottom by 9: 72/36.
4. Add the numerators: 52 + 72 = 124.
5. The resulting fraction is 124/36.
6. Simplify by ... |
Calculate the mean, population variance, and standard deviation for the dataset: [4, 8, 9, 8, 3] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (4 + 8 + 9 + 8 + 3) / 5 = 32 / 5 = 6.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-2.4, 1.6, 2.6, 1.6, -3.4]
Squared Differences: [5.76, 2.56, 6.76, 2.56, 11.56]
Sum ... |
Find the derivative of f(x) = (3x + 3)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 3x + 3.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 3x + 3. Its derivative is h'(x) = 3.
4. Multiply g'(u) by h'(x): f'(x) = 3(3x + 3)^2 * 3 = 9(3x +... |
Find the derivative of f(x) = (5x + 5)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 5x + 5.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 5x + 5. Its derivative is h'(x) = 5.
4. Multiply g'(u) by h'(x): f'(x) = 3(5x + 5)^2 * 5 = 15(5x ... |
In a right-angled triangle, the lengths of the two legs are a = 14 and b = 3. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 14^2 + 3^2 = c^2
2. Calculate the squares: 196 + 9 = c^2
3. Sum the squares: 205 = c^2
4. Take the square root of both sides: c = √205
Final Answ... |
Multiply the following 2x2 matrices:
A =
[1, 5]
[1, 1]
B =
[1, 0]
[5, -4] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (1*1) + (5*5) = 26
C12 = (A11 * B12) + (A12 * B22) = (1*0) + (5*-4) = -20
C21 = (A21 * B11) + (A22 * B21) = (1*1) + (1*5) = 6
C22 = (A21 * B12) + (A22 * B22) = (1*0) + (1*-4) = -4
Result Matr... |
Compute the indefinite integral of f(x) = 6x^1. | To find the indefinite integral ∫ 6x^1 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 1 + 1 = 2.
2. Divide the coefficient by the new exponent: 6 / 2 = 3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 6x^1 dx = 3x^2 + C |
Evaluate tan(0°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of tan(0°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.0000 (exact form depends on root simplification if applicable) |
Evaluate sin(60°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of sin(60°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.8660 (exact form depends on root simplification if applicable) |
Multiply the following 2x2 matrices:
A =
[-4, -3]
[-2, -3]
B =
[0, -1]
[-2, 5] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-4*0) + (-3*-2) = 6
C12 = (A11 * B12) + (A12 * B22) = (-4*-1) + (-3*5) = -11
C21 = (A21 * B11) + (A22 * B21) = (-2*0) + (-3*-2) = 6
C22 = (A21 * B12) + (A22 * B22) = (-2*-1) + (-3*5) = -13
R... |
Calculate the mean, population variance, and standard deviation for the dataset: [16, 17, 5, 19, 16] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (16 + 17 + 5 + 19 + 16) / 5 = 73 / 5 = 14.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [1.4, 2.4, -9.6, 4.4, 1.4]
Squared Differences: [1.96, 5.76, 92.16, 19.36, 1.96]
... |
Multiply the following 2x2 matrices:
A =
[5, 1]
[4, -5]
B =
[2, -2]
[4, 5] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (5*2) + (1*4) = 14
C12 = (A11 * B12) + (A12 * B22) = (5*-2) + (1*5) = -5
C21 = (A21 * B11) + (A22 * B21) = (4*2) + (-5*4) = -12
C22 = (A21 * B12) + (A22 * B22) = (4*-2) + (-5*5) = -33
Result ... |
Find the roots of the quadratic equation: 10x^2 + -2x + 10 = 0 | 1. Identify coefficients: a=10, b=-2, c=10.
2. Calculate the discriminant: Δ = b^2 - 4ac = -2^2 - 4(10)(10) = 4 - 400 = -396.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (2 ± √396i) / 20.
Final Answer: x = 0.1 ± 0.99498743710662i |
Compute the indefinite integral of f(x) = 7x^2. | To find the indefinite integral ∫ 7x^2 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 2 + 1 = 3.
2. Divide the coefficient by the new exponent: 7 / 3 = 7/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 7x^2 dx = 7/3x^3 + C |
In a right-angled triangle, the lengths of the two legs are a = 9 and b = 15. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 9^2 + 15^2 = c^2
2. Calculate the squares: 81 + 225 = c^2
3. Sum the squares: 306 = c^2
4. Take the square root of both sides: c = √306
Final Ans... |
Calculate the mean, population variance, and standard deviation for the dataset: [15, 5, 6, 15, 10] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (15 + 5 + 6 + 15 + 10) / 5 = 51 / 5 = 10.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [4.8, -5.2, -4.2, 4.8, -0.2]
Squared Differences: [23.04, 27.04, 17.64, 23.04, 0.04]... |
Calculate the mean, population variance, and standard deviation for the dataset: [7, 6, 20, 12, 6] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (7 + 6 + 20 + 12 + 6) / 5 = 51 / 5 = 10.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-3.2, -4.2, 9.8, 1.8, -4.2]
Squared Differences: [10.24, 17.64, 96.04, 3.24, 17.64]
... |
Find the derivative of f(x) = (5x + 4)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 5x + 4.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 5x + 4. Its derivative is h'(x) = 5.
4. Multiply g'(u) by h'(x): f'(x) = 3(5x + 4)^2 * 5 = 15(5x ... |
Multiply the following 2x2 matrices:
A =
[-5, 1]
[5, -1]
B =
[3, 4]
[1, 3] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-5*3) + (1*1) = -14
C12 = (A11 * B12) + (A12 * B22) = (-5*4) + (1*3) = -17
C21 = (A21 * B11) + (A22 * B21) = (5*3) + (-1*1) = 14
C22 = (A21 * B12) + (A22 * B22) = (5*4) + (-1*3) = 17
Result ... |
Compute the indefinite integral of f(x) = 10x^4. | To find the indefinite integral ∫ 10x^4 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 4 + 1 = 5.
2. Divide the coefficient by the new exponent: 10 / 5 = 2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 10x^4 dx = 2x^5 + C |
Multiply the following 2x2 matrices:
A =
[-1, 3]
[4, 0]
B =
[4, 2]
[2, 1] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-1*4) + (3*2) = 2
C12 = (A11 * B12) + (A12 * B22) = (-1*2) + (3*1) = 1
C21 = (A21 * B11) + (A22 * B21) = (4*4) + (0*2) = 16
C22 = (A21 * B12) + (A22 * B22) = (4*2) + (0*1) = 8
Result Matrix ... |
In a right-angled triangle, the lengths of the two legs are a = 10 and b = 8. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 10^2 + 8^2 = c^2
2. Calculate the squares: 100 + 64 = c^2
3. Sum the squares: 164 = c^2
4. Take the square root of both sides: c = √164
Final Ans... |
Multiply the following 2x2 matrices:
A =
[-5, -2]
[0, -5]
B =
[5, 3]
[-1, 2] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-5*5) + (-2*-1) = -23
C12 = (A11 * B12) + (A12 * B22) = (-5*3) + (-2*2) = -19
C21 = (A21 * B11) + (A22 * B21) = (0*5) + (-5*-1) = 5
C22 = (A21 * B12) + (A22 * B22) = (0*3) + (-5*2) = -10
Res... |
Calculate the mean, population variance, and standard deviation for the dataset: [14, 2, 6, 5, 16] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (14 + 2 + 6 + 5 + 16) / 5 = 43 / 5 = 8.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [5.4, -6.6, -2.6, -3.6, 7.4]
Squared Differences: [29.16, 43.56, 6.76, 12.96, 54.76]
... |
In a right-angled triangle, the lengths of the two legs are a = 15 and b = 8. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 15^2 + 8^2 = c^2
2. Calculate the squares: 225 + 64 = c^2
3. Sum the squares: 289 = c^2
4. Take the square root of both sides: c = √289
Final Ans... |
In a right-angled triangle, the lengths of the two legs are a = 14 and b = 11. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 14^2 + 11^2 = c^2
2. Calculate the squares: 196 + 121 = c^2
3. Sum the squares: 317 = c^2
4. Take the square root of both sides: c = √317
Final A... |
Add the fractions 8/2 and 14/11. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 2 and 11, which is 22.
2. Convert 8/2 by multiplying top and bottom by 11: 88/22.
3. Convert 14/11 by multiplying top and bottom by 2: 28/22.
4. Add the numerators: 88 + 28 = 116.
5. The resulting fraction is 116/22.
6. Simplify ... |
Find the roots of the quadratic equation: 8x^2 + -10x + -8 = 0 | 1. Identify coefficients: a=8, b=-10, c=-8.
2. Calculate the discriminant: Δ = b^2 - 4ac = 100 - -256 = 356.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (10 ± √356) / 16.
5. Calculate roots: x1 = 1.8042476415070754, x2 = -0.5542476415070754.
Fi... |
Evaluate cos(60°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of cos(60°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.5000 (exact form depends on root simplification if applicable) |
Compute the indefinite integral of f(x) = 9x^4. | To find the indefinite integral ∫ 9x^4 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 4 + 1 = 5.
2. Divide the coefficient by the new exponent: 9 / 5 = 9/5.
3. Combine and add the constant of integration C.
Final Answer: ∫ 9x^4 dx = 9/5x^5 + C |
Compute the quotient and remainder of 92431 divided by 69. Show the long division steps. | To divide 92431 by 69:
1. Determine how many times 69 goes into the leading digits of 92431.
2. Multiply the divisor by this estimate, subtract from the current dividend portion, and bring down the next digit.
3. Repeat until all digits are processed.
Result: 92431 / 69 = 1339 with a remainder of 40.
Proof: (1339 * 69... |
Find the roots of the quadratic equation: 4x^2 + 2x + 4 = 0 | 1. Identify coefficients: a=4, b=2, c=4.
2. Calculate the discriminant: Δ = b^2 - 4ac = 2^2 - 4(4)(4) = 4 - 64 = -60.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-2 ± √60i) / 8.
Final Answer: x = -0.25 ± 0.9682458365518543i |
Compute the indefinite integral of f(x) = 8x^1. | To find the indefinite integral ∫ 8x^1 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 1 + 1 = 2.
2. Divide the coefficient by the new exponent: 8 / 2 = 4.
3. Combine and add the constant of integration C.
Final Answer: ∫ 8x^1 dx = 4x^2 + C |
Evaluate the logarithm: log_2(32) | To evaluate log_2(32), we ask the question: '2 raised to what power equals 32?'
Let x be the unknown power: 2^x = 32
Since 2^5 = 32, it follows that x = 5.
Final Answer: 5 |
Calculate the mean, population variance, and standard deviation for the dataset: [12, 10, 5, 9, 1] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (12 + 10 + 5 + 9 + 1) / 5 = 37 / 5 = 7.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [4.6, 2.6, -2.4, 1.6, -6.4]
Squared Differences: [21.16, 6.76, 5.76, 2.56, 40.96]
S... |
Compute the indefinite integral of f(x) = 3x^4. | To find the indefinite integral ∫ 3x^4 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 4 + 1 = 5.
2. Divide the coefficient by the new exponent: 3 / 5 = 3/5.
3. Combine and add the constant of integration C.
Final Answer: ∫ 3x^4 dx = 3/5x^5 + C |
Evaluate the logarithm: log_5(3125) | To evaluate log_5(3125), we ask the question: '5 raised to what power equals 3125?'
Let x be the unknown power: 5^x = 3125
Since 5^5 = 3125, it follows that x = 5.
Final Answer: 5 |
In a right-angled triangle, the lengths of the two legs are a = 13 and b = 15. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 13^2 + 15^2 = c^2
2. Calculate the squares: 169 + 225 = c^2
3. Sum the squares: 394 = c^2
4. Take the square root of both sides: c = √394
Final A... |
Evaluate tan(45°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of tan(45°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 1.0000 (exact form depends on root simplification if applicable) |
Add the fractions 9/11 and 12/8. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 11 and 8, which is 88.
2. Convert 9/11 by multiplying top and bottom by 8: 72/88.
3. Convert 12/8 by multiplying top and bottom by 11: 132/88.
4. Add the numerators: 72 + 132 = 204.
5. The resulting fraction is 204/88.
6. Simplif... |
Calculate the mean, population variance, and standard deviation for the dataset: [16, 4, 1, 15, 9] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (16 + 4 + 1 + 15 + 9) / 5 = 45 / 5 = 9.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [7.0, -5.0, -8.0, 6.0, 0.0]
Squared Differences: [49.0, 25.0, 64.0, 36.0, 0.0]
Sum ... |
Multiply the following 2x2 matrices:
A =
[4, 3]
[-5, 1]
B =
[4, -4]
[-2, -3] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (4*4) + (3*-2) = 10
C12 = (A11 * B12) + (A12 * B22) = (4*-4) + (3*-3) = -25
C21 = (A21 * B11) + (A22 * B21) = (-5*4) + (1*-2) = -22
C22 = (A21 * B12) + (A22 * B22) = (-5*-4) + (1*-3) = 17
Res... |
In a right-angled triangle, the lengths of the two legs are a = 20 and b = 14. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 20^2 + 14^2 = c^2
2. Calculate the squares: 400 + 196 = c^2
3. Sum the squares: 596 = c^2
4. Take the square root of both sides: c = √596
Final A... |
In a right-angled triangle, the lengths of the two legs are a = 7 and b = 10. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 7^2 + 10^2 = c^2
2. Calculate the squares: 49 + 100 = c^2
3. Sum the squares: 149 = c^2
4. Take the square root of both sides: c = √149
Final Ans... |
Evaluate cos(60°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of cos(60°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.5000 (exact form depends on root simplification if applicable) |
Add the fractions 14/3 and 8/7. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 3 and 7, which is 21.
2. Convert 14/3 by multiplying top and bottom by 7: 98/21.
3. Convert 8/7 by multiplying top and bottom by 3: 24/21.
4. Add the numerators: 98 + 24 = 122.
5. The resulting fraction is 122/21.
Final Answer: 1... |
Find the roots of the quadratic equation: 7x^2 + -6x + -8 = 0 | 1. Identify coefficients: a=7, b=-6, c=-8.
2. Calculate the discriminant: Δ = b^2 - 4ac = 36 - -224 = 260.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (6 ± √260) / 14.
5. Calculate roots: x1 = 1.5803225354712214, x2 = -0.7231796783283642.
Final... |
Find the derivative of f(x) = (5x + 5)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 5x + 5.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 5x + 5. Its derivative is h'(x) = 5.
4. Multiply g'(u) by h'(x): f'(x) = 3(5x + 5)^2 * 5 = 15(5x ... |
Compute the indefinite integral of f(x) = 10x^4. | To find the indefinite integral ∫ 10x^4 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 4 + 1 = 5.
2. Divide the coefficient by the new exponent: 10 / 5 = 2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 10x^4 dx = 2x^5 + C |
In a right-angled triangle, the lengths of the two legs are a = 9 and b = 12. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 9^2 + 12^2 = c^2
2. Calculate the squares: 81 + 144 = c^2
3. Sum the squares: 225 = c^2
4. Take the square root of both sides: c = √225
Final Ans... |
Find the derivative of f(x) = (2x + 3)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 2x + 3.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 2x + 3. Its derivative is h'(x) = 2.
4. Multiply g'(u) by h'(x): f'(x) = 3(2x + 3)^2 * 2 = 6(2x +... |
Compute the indefinite integral of f(x) = 8x^5. | To find the indefinite integral ∫ 8x^5 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 5 + 1 = 6.
2. Divide the coefficient by the new exponent: 8 / 6 = 4/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 8x^5 dx = 4/3x^6 + C |
In a right-angled triangle, the lengths of the two legs are a = 5 and b = 12. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 5^2 + 12^2 = c^2
2. Calculate the squares: 25 + 144 = c^2
3. Sum the squares: 169 = c^2
4. Take the square root of both sides: c = √169
Final Ans... |
Multiply the following 2x2 matrices:
A =
[4, -4]
[-4, -3]
B =
[-5, 3]
[-5, -2] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (4*-5) + (-4*-5) = 0
C12 = (A11 * B12) + (A12 * B22) = (4*3) + (-4*-2) = 20
C21 = (A21 * B11) + (A22 * B21) = (-4*-5) + (-3*-5) = 35
C22 = (A21 * B12) + (A22 * B22) = (-4*3) + (-3*-2) = -6
Re... |
Calculate the mean, population variance, and standard deviation for the dataset: [5, 7, 19, 10, 16] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (5 + 7 + 19 + 10 + 16) / 5 = 57 / 5 = 11.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-6.4, -4.4, 7.6, -1.4, 4.6]
Squared Differences: [40.96, 19.36, 57.76, 1.96, 21.16]... |
Compute the indefinite integral of f(x) = 2x^2. | To find the indefinite integral ∫ 2x^2 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 2 + 1 = 3.
2. Divide the coefficient by the new exponent: 2 / 3 = 2/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 2x^2 dx = 2/3x^3 + C |
Find the derivative of f(x) = (4x + 4)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 4x + 4.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 4x + 4. Its derivative is h'(x) = 4.
4. Multiply g'(u) by h'(x): f'(x) = 3(4x + 4)^2 * 4 = 12(4x ... |
Add the fractions 6/6 and 4/6. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 6 and 6, which is 6.
2. Convert 6/6 by multiplying top and bottom by 1: 6/6.
3. Convert 4/6 by multiplying top and bottom by 1: 4/6.
4. Add the numerators: 6 + 4 = 10.
5. The resulting fraction is 10/6.
6. Simplify by dividing nu... |
Add the fractions 7/12 and 5/4. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 12 and 4, which is 12.
2. Convert 7/12 by multiplying top and bottom by 1: 7/12.
3. Convert 5/4 by multiplying top and bottom by 3: 15/12.
4. Add the numerators: 7 + 15 = 22.
5. The resulting fraction is 22/12.
6. Simplify by div... |
Find the derivative of f(x) = (5x + 3)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 5x + 3.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 5x + 3. Its derivative is h'(x) = 5.
4. Multiply g'(u) by h'(x): f'(x) = 3(5x + 3)^2 * 5 = 15(5x ... |
Compute the quotient and remainder of 21924 divided by 10. Show the long division steps. | To divide 21924 by 10:
1. Determine how many times 10 goes into the leading digits of 21924.
2. Multiply the divisor by this estimate, subtract from the current dividend portion, and bring down the next digit.
3. Repeat until all digits are processed.
Result: 21924 / 10 = 2192 with a remainder of 4.
Proof: (2192 * 10)... |
In a right-angled triangle, the lengths of the two legs are a = 11 and b = 10. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 11^2 + 10^2 = c^2
2. Calculate the squares: 121 + 100 = c^2
3. Sum the squares: 221 = c^2
4. Take the square root of both sides: c = √221
Final A... |
Evaluate the logarithm: log_4(64) | To evaluate log_4(64), we ask the question: '4 raised to what power equals 64?'
Let x be the unknown power: 4^x = 64
Since 4^3 = 64, it follows that x = 3.
Final Answer: 3 |
In a right-angled triangle, the lengths of the two legs are a = 16 and b = 17. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 16^2 + 17^2 = c^2
2. Calculate the squares: 256 + 289 = c^2
3. Sum the squares: 545 = c^2
4. Take the square root of both sides: c = √545
Final A... |
Find the derivative of f(x) = (5x + 3)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 5x + 3.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 5x + 3. Its derivative is h'(x) = 5.
4. Multiply g'(u) by h'(x): f'(x) = 3(5x + 3)^2 * 5 = 15(5x ... |
Compute the quotient and remainder of 74053 divided by 90. Show the long division steps. | To divide 74053 by 90:
1. Determine how many times 90 goes into the leading digits of 74053.
2. Multiply the divisor by this estimate, subtract from the current dividend portion, and bring down the next digit.
3. Repeat until all digits are processed.
Result: 74053 / 90 = 822 with a remainder of 73.
Proof: (822 * 90) ... |
Evaluate the logarithm: log_2(4) | To evaluate log_2(4), we ask the question: '2 raised to what power equals 4?'
Let x be the unknown power: 2^x = 4
Since 2^2 = 4, it follows that x = 2.
Final Answer: 2 |
In a right-angled triangle, the lengths of the two legs are a = 7 and b = 18. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 7^2 + 18^2 = c^2
2. Calculate the squares: 49 + 324 = c^2
3. Sum the squares: 373 = c^2
4. Take the square root of both sides: c = √373
Final Ans... |
Compute the indefinite integral of f(x) = 6x^5. | To find the indefinite integral ∫ 6x^5 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 5 + 1 = 6.
2. Divide the coefficient by the new exponent: 6 / 6 = 1.
3. Combine and add the constant of integration C.
Final Answer: ∫ 6x^5 dx = 1x^6 + C |
Multiply the following 2x2 matrices:
A =
[5, 3]
[3, 0]
B =
[-5, 1]
[1, -1] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (5*-5) + (3*1) = -22
C12 = (A11 * B12) + (A12 * B22) = (5*1) + (3*-1) = 2
C21 = (A21 * B11) + (A22 * B21) = (3*-5) + (0*1) = -15
C22 = (A21 * B12) + (A22 * B22) = (3*1) + (0*-1) = 3
Result Ma... |
Compute the quotient and remainder of 88406 divided by 53. Show the long division steps. | To divide 88406 by 53:
1. Determine how many times 53 goes into the leading digits of 88406.
2. Multiply the divisor by this estimate, subtract from the current dividend portion, and bring down the next digit.
3. Repeat until all digits are processed.
Result: 88406 / 53 = 1668 with a remainder of 2.
Proof: (1668 * 53)... |
Compute the indefinite integral of f(x) = 7x^5. | To find the indefinite integral ∫ 7x^5 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 5 + 1 = 6.
2. Divide the coefficient by the new exponent: 7 / 6 = 7/6.
3. Combine and add the constant of integration C.
Final Answer: ∫ 7x^5 dx = 7/6x^6 + C |
Find the derivative of f(x) = (4x + 4)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 4x + 4.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 4x + 4. Its derivative is h'(x) = 4.
4. Multiply g'(u) by h'(x): f'(x) = 3(4x + 4)^2 * 4 = 12(4x ... |
Find the derivative of f(x) = (2x + 5)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 2x + 5.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 2x + 5. Its derivative is h'(x) = 2.
4. Multiply g'(u) by h'(x): f'(x) = 3(2x + 5)^2 * 2 = 6(2x +... |
Find the derivative of f(x) = (5x + 5)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 5x + 5.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 5x + 5. Its derivative is h'(x) = 5.
4. Multiply g'(u) by h'(x): f'(x) = 3(5x + 5)^2 * 5 = 15(5x ... |
In a right-angled triangle, the lengths of the two legs are a = 17 and b = 19. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 17^2 + 19^2 = c^2
2. Calculate the squares: 289 + 361 = c^2
3. Sum the squares: 650 = c^2
4. Take the square root of both sides: c = √650
Final A... |
Find the roots of the quadratic equation: 2x^2 + 2x + -1 = 0 | 1. Identify coefficients: a=2, b=2, c=-1.
2. Calculate the discriminant: Δ = b^2 - 4ac = 4 - -8 = 12.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-2 ± √12) / 4.
5. Calculate roots: x1 = 0.3660254037844386, x2 = -1.3660254037844386.
Final Answe... |
Add the fractions 6/14 and 5/12. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 14 and 12, which is 84.
2. Convert 6/14 by multiplying top and bottom by 6: 36/84.
3. Convert 5/12 by multiplying top and bottom by 7: 35/84.
4. Add the numerators: 36 + 35 = 71.
5. The resulting fraction is 71/84.
Final Answer: ... |
Evaluate tan(45°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of tan(45°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 1.0000 (exact form depends on root simplification if applicable) |
Calculate the mean, population variance, and standard deviation for the dataset: [15, 19, 13, 13, 16] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (15 + 19 + 13 + 13 + 16) / 5 = 76 / 5 = 15.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-0.2, 3.8, -2.2, -2.2, 0.8]
Squared Differences: [0.04, 14.44, 4.84, 4.84, 0.64]
... |
Multiply the following 2x2 matrices:
A =
[-1, -4]
[2, 0]
B =
[-4, 5]
[4, 4] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-1*-4) + (-4*4) = -12
C12 = (A11 * B12) + (A12 * B22) = (-1*5) + (-4*4) = -21
C21 = (A21 * B11) + (A22 * B21) = (2*-4) + (0*4) = -8
C22 = (A21 * B12) + (A22 * B22) = (2*5) + (0*4) = 10
Resul... |
Find the derivative of f(x) = (2x + 5)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 2x + 5.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 2x + 5. Its derivative is h'(x) = 2.
4. Multiply g'(u) by h'(x): f'(x) = 3(2x + 5)^2 * 2 = 6(2x +... |
Find the derivative of f(x) = (4x + 2)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 4x + 2.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 4x + 2. Its derivative is h'(x) = 4.
4. Multiply g'(u) by h'(x): f'(x) = 3(4x + 2)^2 * 4 = 12(4x ... |
Find the derivative of f(x) = (3x + 5)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 3x + 5.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 3x + 5. Its derivative is h'(x) = 3.
4. Multiply g'(u) by h'(x): f'(x) = 3(3x + 5)^2 * 3 = 9(3x +... |
Calculate the mean, population variance, and standard deviation for the dataset: [17, 19, 7, 11, 18] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (17 + 19 + 7 + 11 + 18) / 5 = 72 / 5 = 14.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [2.6, 4.6, -7.4, -3.4, 3.6]
Squared Differences: [6.76, 21.16, 54.76, 11.56, 12.96]... |
Compute the indefinite integral of f(x) = 7x^2. | To find the indefinite integral ∫ 7x^2 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 2 + 1 = 3.
2. Divide the coefficient by the new exponent: 7 / 3 = 7/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 7x^2 dx = 7/3x^3 + C |
Add the fractions 9/9 and 15/14. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 9 and 14, which is 126.
2. Convert 9/9 by multiplying top and bottom by 14: 126/126.
3. Convert 15/14 by multiplying top and bottom by 9: 135/126.
4. Add the numerators: 126 + 135 = 261.
5. The resulting fraction is 261/126.
6. S... |
In a right-angled triangle, the lengths of the two legs are a = 16 and b = 11. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 16^2 + 11^2 = c^2
2. Calculate the squares: 256 + 121 = c^2
3. Sum the squares: 377 = c^2
4. Take the square root of both sides: c = √377
Final A... |
Add the fractions 4/4 and 1/10. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 4 and 10, which is 20.
2. Convert 4/4 by multiplying top and bottom by 5: 20/20.
3. Convert 1/10 by multiplying top and bottom by 2: 2/20.
4. Add the numerators: 20 + 2 = 22.
5. The resulting fraction is 22/20.
6. Simplify by div... |
Multiply the following 2x2 matrices:
A =
[-3, 4]
[2, -2]
B =
[2, 5]
[0, -3] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-3*2) + (4*0) = -6
C12 = (A11 * B12) + (A12 * B22) = (-3*5) + (4*-3) = -27
C21 = (A21 * B11) + (A22 * B21) = (2*2) + (-2*0) = 4
C22 = (A21 * B12) + (A22 * B22) = (2*5) + (-2*-3) = 16
Result ... |
Add the fractions 5/9 and 3/5. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 9 and 5, which is 45.
2. Convert 5/9 by multiplying top and bottom by 5: 25/45.
3. Convert 3/5 by multiplying top and bottom by 9: 27/45.
4. Add the numerators: 25 + 27 = 52.
5. The resulting fraction is 52/45.
Final Answer: 52/4... |
In a right-angled triangle, the lengths of the two legs are a = 12 and b = 9. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 12^2 + 9^2 = c^2
2. Calculate the squares: 144 + 81 = c^2
3. Sum the squares: 225 = c^2
4. Take the square root of both sides: c = √225
Final Ans... |
Find the roots of the quadratic equation: 1x^2 + 1x + 6 = 0 | 1. Identify coefficients: a=1, b=1, c=6.
2. Calculate the discriminant: Δ = b^2 - 4ac = 1^2 - 4(1)(6) = 1 - 24 = -23.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-1 ± √23i) / 2.
Final Answer: x = -0.5 ± 2.3979157616563596i |
Find the derivative of f(x) = (4x + 5)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 4x + 5.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 4x + 5. Its derivative is h'(x) = 4.
4. Multiply g'(u) by h'(x): f'(x) = 3(4x + 5)^2 * 4 = 12(4x ... |
Compute the indefinite integral of f(x) = 10x^5. | To find the indefinite integral ∫ 10x^5 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 5 + 1 = 6.
2. Divide the coefficient by the new exponent: 10 / 6 = 5/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 10x^5 dx = 5/3x^6 + C |
Find the derivative of f(x) = (4x + 5)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 4x + 5.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 4x + 5. Its derivative is h'(x) = 4.
4. Multiply g'(u) by h'(x): f'(x) = 3(4x + 5)^2 * 4 = 12(4x ... |
Compute the indefinite integral of f(x) = 5x^3. | To find the indefinite integral ∫ 5x^3 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 3 + 1 = 4.
2. Divide the coefficient by the new exponent: 5 / 4 = 5/4.
3. Combine and add the constant of integration C.
Final Answer: ∫ 5x^3 dx = 5/4x^4 + C |
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