Practice Questions

Identify the term used for a point c in the domain of a function f where either f'(c) = 0 or f is not differentiable at c.

The total cost function for producing $x$ units is $C(x) = 100 + 50x - 0.5x^2$. Calculate the marginal cost when 10 units are produced.

Justify why the function f(x) = \log(x) has no local maxima or minima on its domain (0, \infty).

The side of a square is increasing at the rate of $0.2 \text{ cm/s}$. Calculate the rate of increase of its perimeter.

Explain the relationship between the sign of the first derivative of a function, $f'(x)$, and the increasing or decreasing nature of the function $f(x)$.

Define the term 'marginal cost' in the context of economics.

Evaluate the approximate change in the volume of a sphere when its radius increases from 10 cm to 10.02 cm. Propose a formula for this approximation using differentials.

Analyze if the function $f(x) = 1 - 6x - 3x^2$ is increasing or decreasing at $x = -2$.

Justify the statement: The rate of change of the volume of a sphere with respect to its radius is numerically equal to its surface area.

State the condition, using the first derivative, for a function f(x) to be strictly decreasing on an open interval (a, b).

The total revenue $R(x)$ in Rupees from the sale of $x$ items is given by $R(x) = 10x^2 + 20x + 5$. Explain how to find the marginal revenue and describe what this value represents.

Explain the concept of 'rate of change of quantities' using derivatives. Illustrate with the example of a melting snowball (a sphere), explaining how the rate of change of its surface area is related to the rate of change of its radius.

A student claims that if f'(c) = 0 and f''(c) > 0, then c is a point of absolute minima for the function f on any closed interval [a, b] containing c. Critique this claim.

Calculate the equation of the tangent to the curve $y = x^3 - x + 1$ at the point where $x=1$.

Find the critical points of the function $f(x) = x^3 - 12x + 5$.

A function $f(x)$ has a critical point at $x=c$. If $f'(c) = 0$ and $f''(c) = -5$, examine if $x=c$ is a point of local maximum or local minimum.

The radius of a spherical soap bubble is decreasing at the rate of $0.5 \text{ cm/s}$. At what rate is its volume decreasing when the radius is $2 \text{ cm}$?

Find the intervals in which the function $f(x) = 2x^3 - 9x^2 + 12x + 15$ is strictly increasing or decreasing.

Summarize the complete procedure for finding and classifying all local extrema (local maxima and local minima) for a given differentiable function $f(x)$. Your summary should detail how to find critical points and how to use both the First and Second Derivative Tests.

A student finds that for f(x) = (x-2)^4, f'(2) = 0 and f''(2) = 0. They conclude the second derivative test fails and stop. Critique this approach and propose the necessary next step.

Prove that for x > 0, the function f(x) = x + \frac{1}{x} has a local minimum. Then, justify that this local minimum value is 2.

According to the Second Derivative Test, what are the two conditions that must be met for a point $x = c$ to be a point of local maxima for a function $f$?

Explain the difference between a 'local minimum' and an 'absolute minimum' for a function defined on a closed interval $[a, b]$.

Describe the First Derivative Test for determining if a critical point $c$ is a point of local minima.

Recall the formula for the rate of change of y with respect to x, given that two variables x and y are varying with respect to a third variable t.

Summarize the steps required to find the absolute maximum and absolute minimum values of a continuous function $f(x)$ on a closed interval $[a, b]$.

Find the local maximum and local minimum values of the function $f(x) = x^3 - 6x^2 + 9x + 1$.

Find the absolute maximum and minimum values of the function $f(x) = x^3 - 3x^2 + 5$ on the interval $[-1, 3]$.

Two cars start from the same point at the same time. Car A travels north at 60 km/h and Car B travels east at 80 km/h. Formulate an expression for the rate at which the distance between the cars is increasing after 2 hours and evaluate it.

Prove that the rectangle of maximum area that can be inscribed in a circle of radius R is a square. Justify each step of the derivation, including setting up the function and using a derivative test to confirm the maximum.

A particle moves along the curve y = x^3 + 2x. At what point(s) on the curve is the y-coordinate changing twice as fast as the x-coordinate? Justify your answer.

Demonstrate that the function $f(x) = \log(\sin x)$ is strictly increasing on the interval $(0, \frac{\pi}{2})$.

Show that the altitude of the right circular cylinder of maximum volume that can be inscribed in a given right circular cone of height $H$ and radius $R$ is one-third the height of the cone.

Design a container in the shape of a right circular cylinder with an open top that has a fixed surface area of 100\pi cm². Formulate the volume V as a function of the radius r. Determine the dimensions (radius and height) that will maximize the volume, and justify your result using the second derivative test.

Sand is being poured from a pipe at the rate of $16 \text{ cm}^3/\text{s}$. The falling sand forms a cone on the ground in such a way that the diameter of the base is always equal to its height. How fast is the height of the sand cone increasing when the height is $4 \text{ cm}$?

A window is designed in the form of a rectangle surmounted by a semicircle. The total perimeter of the window is 10 m. Formulate an expression for the area of the window as a function of the radius of the semicircle. Propose the dimensions of the window that will admit the maximum possible light and justify your proposal.

A wire of length 20 m is to be cut into two pieces. One piece will be bent to form a square and the other to form an equilateral triangle. What should be the lengths of the two pieces so that the combined area of the two is minimum?

Formulate a condition using the signs of the first derivative, f'(x), to identify a point of inflection c where f'(c) = 0.

An airplane is flying at a constant altitude of 4 km and a constant speed of 720 km/h. It passes directly over a radar station. Formulate an equation relating the horizontal distance of the plane from the station and the direct distance between them. Determine the rate at which the distance from the plane to the station is increasing when it is 5 km away from the station.

If a function $f(x)$ has a derivative $f'(x) = x^2$, explain whether the point $x = 0$ is a local maximum, local minimum, or neither, using the First Derivative Test.

Describe the concept of a point of inflection. Explain its relationship to the first and second derivatives of a function. How does the First Derivative Test identify a point of inflection?

A ladder 13 m long is leaning against a vertical wall. The bottom of the ladder is pulled along the ground, away from the wall, at the rate of 2 m/s. Formulate a method to find how fast the angle between the ladder and the ground is decreasing when the foot of the ladder is 5 m away from the wall.

Justify the intervals where the function f(x) = \sin(x) - \cos(x) is strictly increasing or decreasing on the interval (0, 2\pi).

Describe a scenario in which the Second Derivative Test is inconclusive or fails, and explain the subsequent procedure to classify the critical point.