7. if \\( \\frac{d y}{d x}=16 \\sin ^{3} x \\cos x \\) and \\( y=16 \\) when \\( x=\\frac{\\pi}{2} \\), what…

7. if \\( \\frac{d y}{d x}=16 \\sin ^{3} x \\cos x \\) and \\( y=16 \\) when \\( x=\\frac{\\pi}{2} \\), what is the value of \\( y \\) when \\( x=\\frac{\\pi}{6} \\) ?\n(a) \\( \\frac{49}{4} \\)\n(b) \\( \\frac{1}{4} \\)\n(c) 10\n(d) 13\n8. if \\( \\mathrm{dy} / \\mathrm{dx}=2 \\mathrm{y}^{2} \\) and if \\( y=-1 \\) when \\( x=1 \\), then when \\( x=2, y= \\)\n(a) \\( -2 / 3 \\)\n(b) \\( -1 / 3 \\)\n(c) 0\n(d) \\( 1 / 3 \\)\n(e) \\( 2 / 3 \\)\n9. if \\( \\frac{d y}{d x}=4 y \\) and if \\( y=4 \\) when \\( x=0 \\), then \\( y= \\)\n(a) \\( 4 e^{4 x} \\)\n(b) \\( e^{4 x} \\)\n(c) \\( 3+e^{4 x} \\)\n(d) \\( 4+e^{4 x} \\)\n(e) \\( 2 x^{2}+4 \\)

7. if \\( \\frac{d y}{d x}=16 \\sin ^{3} x \\cos x \\) and \\( y=16 \\) when \\( x=\\frac{\\pi}{2} \\), what is the value of \\( y \\) when \\( x=\\frac{\\pi}{6} \\) ?\n(a) \\( \\frac{49}{4} \\)\n(b) \\( \\frac{1}{4} \\)\n(c) 10\n(d) 13\n8. if \\( \\mathrm{dy} / \\mathrm{dx}=2 \\mathrm{y}^{2} \\) and if \\( y=-1 \\) when \\( x=1 \\), then when \\( x=2, y= \\)\n(a) \\( -2 / 3 \\)\n(b) \\( -1 / 3 \\)\n(c) 0\n(d) \\( 1 / 3 \\)\n(e) \\( 2 / 3 \\)\n9. if \\( \\frac{d y}{d x}=4 y \\) and if \\( y=4 \\) when \\( x=0 \\), then \\( y= \\)\n(a) \\( 4 e^{4 x} \\)\n(b) \\( e^{4 x} \\)\n(c) \\( 3+e^{4 x} \\)\n(d) \\( 4+e^{4 x} \\)\n(e) \\( 2 x^{2}+4 \\)

Answer

Problem 7

Explanation:

Step1: Integrate the derivative

We know that (y=\int 16\sin^{3}x\cos xdx). Let (u = \sin x), then (du=\cos xdx). So (y = 16\int u^{3}du). Using the power - rule (\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C(n\neq - 1)), we have (y=16\times\frac{u^{4}}{4}+C = 4\sin^{4}x+C).

Step2: Find the constant (C)

Since (y = 16) when (x=\frac{\pi}{2}), substitute (x=\frac{\pi}{2}) and (y = 16) into (y = 4\sin^{4}x+C). When (x=\frac{\pi}{2}), (\sin x=1). Then (16=4\times1^{4}+C), so (C = 12).

Step3: Calculate (y) at (x=\frac{\pi}{6})

Substitute (x=\frac{\pi}{6}) into (y = 4\sin^{4}x+12). When (x=\frac{\pi}{6}), (\sin x=\frac{1}{2}). (y=4\times(\frac{1}{2})^{4}+12=4\times\frac{1}{16}+12=\frac{1}{4}+12=\frac{1 + 48}{4}=\frac{49}{4})

Answer:

A. (\frac{49}{4})

Problem 8

Explanation:

Step1: Separate the variables

Given (\frac{dy}{dx}=2y^{2}), we can rewrite it as (\frac{dy}{y^{2}}=2dx).

Step2: Integrate both sides

Integrate (\int y^{- 2}dy=\int 2dx). Using the power - rule (\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C(n\neq - 1)), we get (-\frac{1}{y}=2x + C).

Step3: Find the constant (C)

Since (y=-1) when (x = 1), substitute (x = 1) and (y=-1) into (-\frac{1}{y}=2x + C). (-\frac{1}{-1}=2\times1+C), so (1 = 2 + C), and (C=-1).

Step4: Solve for (y) when (x = 2)

The equation is (-\frac{1}{y}=2x-1). When (x = 2), (-\frac{1}{y}=2\times2-1=3), then (y=-\frac{1}{3})

Answer:

B. (-\frac{1}{3})

Problem 9

Explanation:

Step1: Separate the variables

Given (\frac{dy}{dx}=4y), we can rewrite it as (\frac{dy}{y}=4dx).

Step2: Integrate both sides

Integrate (\int\frac{dy}{y}=\int 4dx). We know that (\int\frac{1}{y}dy=\ln|y|) and (\int 4dx=4x + C), so (\ln|y|=4x + C). Exponentiating both sides gives (y = Ce^{4x}).

Step3: Find the constant (C)

Since (y = 4) when (x = 0), substitute (x = 0) and (y = 4) into (y = Ce^{4x}). (4 = C\times e^{0}), so (C = 4). The solution is (y = 4e^{4x})

Answer:

A. (4e^{4x})