bc unit 7 (ab topics) quiz ver a\n(a) $\frac{5}{21}$\n(b) $sqrt{33}$\n(c) $4 + e^{4}$\n(d) $5e^{4}$\n7. if…

bc unit 7 (ab topics) quiz ver a\n(a) $\frac{5}{21}$\n(b) $sqrt{33}$\n(c) $4 + e^{4}$\n(d) $5e^{4}$\n7. if $\frac{dy}{dx}=16\\sin^{3}x\\cos x$ 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 $dy/dx = 2y^{2}$ 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 $\frac{dy}{dx}=4y$ and if $y = 4$ when $x = 0$, then $y=$\n(a) $4e^{4x}$\n(b) $e^{4x}$\n(c) $3 + e^{4x}$\n(d) $4 + e^{4x}$\n(e) $2x^{2}+4$

bc unit 7 (ab topics) quiz ver a\n(a) $\frac{5}{21}$\n(b) $sqrt{33}$\n(c) $4 + e^{4}$\n(d) $5e^{4}$\n7. if $\frac{dy}{dx}=16\\sin^{3}x\\cos x$ 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 $dy/dx = 2y^{2}$ 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 $\frac{dy}{dx}=4y$ and if $y = 4$ when $x = 0$, then $y=$\n(a) $4e^{4x}$\n(b) $e^{4x}$\n(c) $3 + e^{4x}$\n(d) $4 + e^{4x}$\n(e) $2x^{2}+4$

Answer

Step - by - Step Format:

Step 1: Solve the differential equation (\frac{dy}{dx}=4y)

Separate the variables: (\frac{dy}{y} = 4dx) Integrate both sides: (\int\frac{dy}{y}=\int4dx) Using the integral formulas (\int\frac{1}{y}dy=\ln|y|+C_1) and (\int4dx = 4x + C_2), we get (\ln|y|=4x + C) (where (C = C_2 - C_1)) Exponentiate both sides: (y = e^{4x + C}=e^{C}e^{4x})

Step 2: Use the initial condition (y = 4) when (x = 0)

Substitute (x = 0) and (y = 4) into (y=e^{C}e^{4x}) (4=e^{C}e^{0}) Since (e^{0}=1), then (e^{C}=4)

Step 3: Write the particular solution

The particular solution is (y = 4e^{4x})

Answer:

A. (4e^{4x})