Contextual Applications of Differentiation

$100\pi \text{ cm}^3/\text{min}$

$30\text{ cm}^3/\text{min}$

$900\text{ cm}^3/\text{min}$

$300\text{ cm}^3/\text{min}$

The volume quadruples.

The volume remains the same.

The volume doubles.

The volume triples.

$V = \frac{1}{3} \pi r^2 h$

$V = \frac{4}{3} \pi r^2 h$

$V = \pi r^2 h$

$V = \pi r^2$

$-\left(\dfrac{\sqrt{(64)}}{\sqrt{(100)}}\right)(0.5)$ m/min

$-(10)\cos^{-1}(60^\circ)(0)$ m/min

$-\left(\dfrac{\sqrt{(36)}}{\sqrt{(100)}}\right)(0)$ m/min

$-\left(\dfrac{\sqrt{(84)}}{\sqrt{(100)}}\right)(0)$ m/min

Algebraic methods focus on static measurements whereas calculus addresses dynamic situations where variables depend on each other's rates of change.

Differential calculus allows us to relate instantaneous rates of change between two or more quantities in motion.

Only differential calculus can account for both constant and non-constant speeds within one framework effectively.

Algebraic methods alone cannot provide information about immediate changes in distance at specific instances in time.

The volume doubles.

The volume quadruples.

The volume triples.

The volume remains the same.

To 2 decimal places

To 3 decimal places

To the nearest tenth

To the nearest whole number

Trigonometric methods inherently involve complex calculations, rendering comparisons unnecessary.

Geometric approaches assume angles remain constant, thus precluding the need for differential comparison.

Comparing differentials helps identify which variable’s rate influences others more straightforwardly, simplifying solution processes.

Differentials don’t affect choice since both trigonometry and geometry deal equally well with varying arc lengths.

The shadow does not shrink.

$-0.8$m/s

$0$ m/s

$-1.6$m/s

Height

Radius

All of the above

Slant height