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# Tutor profile: Harish J.

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Harish J.
Calculus, Algebra, other Maths Tutor for 10 years
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## Questions

### Subject:Pre-Calculus

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Question:

What is the equation of the line tangent to the curve $$y = 3e^{2x}$$ and perpendicular to the line $$y = -2x + 10$$

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Harish J.

The slope of line $$y = -2x + 10$$ is $$m_1 = -2$$. It is given that the tangent line is perpendicular to this line. Hence, if $$m_2$$ is the slope of the tangent line, then we have: $$m_1 m_2 = -1$$ or $$-2 m_2 = -1$$ or $$m_2 = \frac{1}{2}$$ Hence, the tangent line belongs to the family of lines: $$y = \frac{1}{2}x + b$$ Now, a tangent to the curve $$y = 3e^{2x}$$ at a general point $$(x,y)$$ would have slope = $$\frac{dy}{dx} = 6e^{2x}$$. Hence, $$6e^{2x} = \frac{1}{2}$$ or $$e^{2x} = \frac{1}{12}$$ or $$x = -1.24245$$ Hence, the tangent line intersects the curve at $$x = -1.24245, y = 3e^{2x} = 0.25$$. Its slope is $$m_2 = \frac{1}{2}$$. Hence, the required equation of the tangent line is: $$y - 0.25 = \frac{1}{2} (x - (-1.24245))$$ or $$y = \frac{1}{2}x + 0.871$$

### Subject:Calculus

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Question:

Find the derivative $$\frac{\mathrm{d}y}{\mathrm{d}x}$$: $$y^3 + x^3 = 3(y-x)$$

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Harish J.

Using implicit differentiation, we have: $$3y^2\frac{\mathrm{d}y}{\mathrm{d}x} + 3x^2 = 3( \frac{\mathrm{d}y}{\mathrm{d}x} - 1)$$ or $$(3y^2-3) \frac{\mathrm{d}y}{\mathrm{d}x} = -x^3 -x$$ or $$\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{x^3 + x}{3-3y^2}$$

### Subject:Algebra

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Question:

Find the equation of a polynomial with roots 2 and -5i, real coefficients, and passing through the point (1, 52).

Inactive
Harish J.

The polynomial has a complex root x = -5i. Hence, it also has the conjugate root x = +5i, so that it would have real coefficients. The root x = 2 is also specified making a total of minimum 3 roots. So, we can choose a cubic polynomial from the family: $$f(x) = k(x - 2)(x - (-5i))(x - 5i)$$ or $$f(x) = k(x - 2)(x^2+25)$$ It is given that it passes through (1, 52). Hence, we have: $$52 = k(1-2)(1^2+25)$$ or $$k = -2$$ Hence, the required polynomial is: $$f(x) = -2(x-2)(x^2+25)$$ or $$f(x) = -2x^3+4x^2-50x+100$$

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