# Tutor profile: Alice P.

## Questions

### Subject: Calculus

Find the derivative of $$f(x)=6x^3−9x+4$$

Differentiating powers removes one from the power. In this question, the power of 3 becomes a power of 2 and the 3 moves to in front of the x and multiplies the 6 to make 18. For the second term, the power of x is 1 and so the x is removed and the 9 remains the same. The third term has no x (or x to the power of zero) and so disappears entirely. The final answer is $$f′(x)=18x^2−9$$

### Subject: Physics

A roller coaster car is raised to a height of $$65$$m and released from rest. What is the maximum possible speed of the car?

At $$65$$m the car only has gravitational potential energy (mass $$\times$$ gravity $$\times$$ height). It will be going at its maximum speed when all of the gravitational potential energy is completely converted to kinetic energy ($$\frac{1}{2} \times$$ mass $$\times$$ velocity$$^2$$) at a height of 0m. By equating these equations and removing the unknown mass variable which is present in both, we can calculate the maximum speed. g $$\times$$ h $$=$$ $$\frac{1}{2} \times$$ v$$^2$$ where g is the gravitational acceleration of Earth $$9.81$$ms$$^{-2}$$, h is the height of $$65$$m, and v is the maximum velocity. Rearranging for v and inputting the numbers, we can determine the maximum velocity of the car is $$35.7$$ms$$^{1}$$

### Subject: Algebra

Solve $$2x^2 + 4x - 16 = 0$$ for $$x$$

Using the quadratic formula $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ with $$a =2$$, $$b=4$$, and $$c=-16$$ we get $$x = -4$$ or $$x = 2$$

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