Applications of Derivatives Review

Front

What is the formula for the area of a circle in terms of its radius?

Back

A = πr²

Front

If the radius of a circle is decreasing at a constant rate, how does it affect the area?

Back

The area decreases at a rate proportional to the circumference.

Front

What is the relationship between the circumference and the radius of a circle?

Back

C = 2πr

Front

How do you find the rate of change of the area of a circle when the radius is changing?

Back

Use the chain rule: dA/dt = (dA/dr)(dr/dt).

Front

What is the significance of the first derivative of a function?

Back

It represents the rate of change or the slope of the function.

Front

What does it mean if the second derivative of a function is positive?

Back

The function is concave up, indicating that the slope is increasing.

Front

How do you determine if a linear approximation is an underestimate or overestimate?

Back

If f''(x) > 0, the linear approximation is an underestimate; if f''(x) < 0, it is an overestimate.

Front

What is the formula for the volume of a rectangular solid with a square base?

Back

V = x²h, where x is the length of the base and h is the height.

Front

How do you find the maximum volume of a solid given a surface area constraint?

Back

Use optimization techniques, setting up equations for volume and surface area.

Front

What is the tangent line approximation formula?

Back

f(a) + f'(a)(x - a) gives the approximation of f(x) near x = a.