Multiply the numbers (2*2=4)
![=\frac{-7+\sqrt{7^2-4\cdot \:2\left(-5\right)}}{2\cdot \:2}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B-7%2B%5Csqrt%7B7%5E2-4%5Ccdot%20%5C%3A2%5Cleft%28-5%5Cright%29%7D%7D%7B2%5Ccdot%20%5C%3A2%7D)
Apply the rule
![\sqrt{7^2+2\cdot \:4\cdot \:5}](https://tex.z-dn.net/?f=%5Csqrt%7B7%5E2%2B2%5Ccdot%20%5C%3A4%5Ccdot%20%5C%3A5%7D)
7^2 =49
![=\sqrt{49+2\cdot \:4\cdot \:5}](https://tex.z-dn.net/?f=%3D%5Csqrt%7B49%2B2%5Ccdot%20%5C%3A4%5Ccdot%20%5C%3A5%7D)
Multiply
![\sqrt{49+40}](https://tex.z-dn.net/?f=%5Csqrt%7B49%2B40%7D)
Add
![\sqrt{89}](https://tex.z-dn.net/?f=%5Csqrt%7B89%7D)
![=\frac{-7-\sqrt{89}}{4}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B-7-%5Csqrt%7B89%7D%7D%7B4%7D)
Final solution:
Answer: See below
Step-by-step explanation:
A.
Let's split the integral into two parts, by the Sum Rule.
[split into 2 integrals]
[solve integral for each part]
[Remember, we need to add C for constant]
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B.
[expand into 2 integrals]
[simplify second integral]
[solve integral for each part]
![2\sqrt{x} +\frac{2}{3}x^3^/^2+C](https://tex.z-dn.net/?f=2%5Csqrt%7Bx%7D%20%2B%5Cfrac%7B2%7D%7B3%7Dx%5E3%5E%2F%5E2%2BC)
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C.
[distribute]
[split into 2 integrals]
[solve integral for each part]
[solve]
![\frac{112}{15}](https://tex.z-dn.net/?f=%5Cfrac%7B112%7D%7B15%7D)
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D. *Note: I can't put -1 for the interval, but know that the 1 on the bottom is supposed to be -1.
[expand]
[split into 2 integrals]
[solve integral for each part]
[solve]
![\frac{4}{3}](https://tex.z-dn.net/?f=%5Cfrac%7B4%7D%7B3%7D)
A) There are a number of ways to compute the determinant of a 3x3 matrix. Since k is on the bottom row, it is convenient to compute the cofactors of the numbers on the bottom row. Then the determinant is ...
1×(2×-1 -3×1) -k×(3×-1 -2×1) +2×(3×3 -2×2) = 5 -5k
bi) Π₁ can be written using r = (x, y, z).
Π₁ ⇒ 3x +2y +z = 4
bii) The cross product of the coefficients of λ and μ will give the normal to the plane. The dot-product of that with the constant vector will give the desired constant.
Π₂ ⇒ ((1, 0, 2)×(1, -1, -1))•(x, y, z) = ((1, 0, 2)×(1, -1, -1))•(1, 2, 3)
Π₂ ⇒ 2x +3y -z = 5
c) If the three planes form a sheath, the ranks of their coefficient matrix and that of the augmented matrix must be 2. That is, the determinant must be zero. The value of k that makes the determinant zero is found in part (a) to be -1.
A common approach to determining the rank of a matrix is to reduce it to row echelon form. Then the number of independent rows becomes obvious. (It is the number of non-zero rows.) This form for k=-1 is shown in the picture.
The answer to that would be 1.04166667.
Answer:
triangle = 8
the rest is mind abuse
Step-by-step explanation: