1.
The rate of change of distance ($s$) with respect to time ($t$) is called _________________.
Answer: Velocity (or Speed in a scalar context).
2.
If $y = f(x)$, then the instantaneous rate of change of $y$ with respect to $x$ is denoted mathematically as _________________.
Answer: $\frac{dy}{dx}$ or $f'(x)$.
3.
Find the expression for the rate of change of the area of a circle ($A = \pi r^2$) with respect to its radius $r$.
Answer: $\frac{dA}{dr} = 2\pi r$.
4.
What is the derivative of the volume of a cube ($V = x^3$) with respect to its edge $x$?
Answer: $\frac{dV}{dx} = 3x^2$.
5.
If a circle's radius $r$ is increasing at $2\text{ cm/s}$, what is the value of $\frac{dr}{dt}$?
Answer: $+2\text{ cm/s}$.
6.
If the volume $V$ of a balloon is decreasing at $5\text{ cm}^3\text{/s}$, what is the value of $\frac{dV}{dt}$?
Answer: $-5\text{ cm}^3\text{/s}$.
7.
According to the chain rule for related rates, $\frac{dy}{dt} = \frac{dy}{dx} \times $ _________________.
Answer: $\frac{dx}{dt}$.
8.
Find the rate of change of the circumference of a circle ($C = 2\pi r$) with respect to $r$.
Answer: $\frac{dC}{dr} = 2\pi$.
9.
True or False: Instantaneous rate of change is measured using derivatives.
Answer: True.
10.
If $y = 3x^2$, find the value of $\frac{dy}{dx}$ at $x = 2$.
Answer: $\frac{dy}{dx} = 6x$. At $x=2$, $6(2) = 12$.
11.
If the side of a square is $a$, what is the expression for the rate of change of its perimeter $P$ with respect to $a$?
Answer: Since $P = 4a$, $\frac{dP}{da} = 4$.
12.
Write the formula for the surface area $S$ of a sphere in terms of its radius $r$.
Answer: $S = 4\pi r^2$.
13.
Find the derivative of the surface area of a sphere ($S = 4\pi r^2$) with respect to $r$.
Answer: $\frac{dS}{dr} = 8\pi r$.
14.
If $x$ and $y$ are related by the equation $y = x^2$, express $\frac{dy}{dt}$ in terms of $x$ and $\frac{dx}{dt}$.
Answer: $\frac{dy}{dt} = 2x \frac{dx}{dt}$.
15.
A continuous function $f(x)$ is strictly increasing on an open interval if $f'(x)$ is _________________ than zero.
Answer: Greater ($>$)
16.
A continuous function $f(x)$ is strictly decreasing on an open interval if $f'(x)$ is _________________ than zero.
Answer: Less ($<$)
17.
If $f'(x) \ge 0$ for all $x$ in an interval, the function is simply called _________________.
Answer: Non-decreasing (or Increasing).
18.
If $f'(x) \le 0$ for all $x$ in an interval, the function is simply called _________________.
Answer: Non-increasing (or Decreasing).
19.
Find $f'(x)$ for $f(x) = 5x + 3$. Is the function strictly increasing or decreasing?
Answer: $f'(x) = 5$. Since $5 > 0$, it is strictly increasing.
20.
Find $f'(x)$ for $f(x) = -2x + 7$. Is the function strictly increasing or decreasing?
Answer: $f'(x) = -2$. Since $-2 < 0$, it is strictly decreasing.
21.
The critical points (or stationary points) for finding intervals of monotonicity are found by setting $f'(x) =$ _________________.
Answer: $0$.
22.
True or False: The exponential function $f(x) = e^x$ is a strictly increasing function for all real numbers.
Answer: True (since $f'(x) = e^x > 0$ always).
23.
What is the first derivative of $f(x) = \sin x$?
Answer: $f'(x) = \cos x$.
24.
On the interval $(0, \pi/2)$, is the function $f(x) = \sin x$ strictly increasing or decreasing?
Answer: Strictly increasing (because $\cos x > 0$ in the first quadrant).
25.
Find the first derivative of $f(x) = x^2$.
Answer: $f'(x) = 2x$.
26.
For the interval $x > 0$, is the function $f(x) = x^2$ strictly increasing or strictly decreasing?
Answer: Strictly increasing (since $2x > 0$ for $x > 0$).
27.
What popular graphical method uses a number line with alternating plus/minus signs to determine intervals of increase/decrease?
Answer: Wavy curve method (or sign chart method).
28.
If $f'(x) = (x-1)(x-2)$, what are the critical points where $f'(x) = 0$?
Answer: $x = 1$ and $x = 2$.
29.
The slope of the tangent to the curve $y = f(x)$ at the point $(x_1, y_1)$ is denoted by the derivative value _________________ evaluated at that point.
Answer: $\frac{dy}{dx}$ (or $f'(x_1)$).
30.
If $m_1$ is the slope of the tangent and $m_2$ is the slope of the normal, what is the relation between $m_1$ and $m_2$?
Answer: $m_1 \times m_2 = -1$.
31.
The slope of the normal to the curve $y = f(x)$ is given by $-1 / $ _________________.
Answer: $\left(\frac{dy}{dx}\right)$ or Slope of Tangent.
32.
Find the slope of the tangent to the curve $y = x^2$ at the point where $x = 1$.
Answer: $\frac{dy}{dx} = 2x$. At $x=1$, slope $= 2$.
33.
Using the result from the previous question, find the slope of the normal to the curve $y = x^2$ at $x = 1$.
Answer: $-1/2$.
34.
Write the point-slope form equation of a straight line passing through point $(x_1, y_1)$ with slope $m$.
Answer: $y - y_1 = m(x - x_1)$.
35.
Write the standard equation of the tangent to the curve $y=f(x)$ at the point $(x_0, y_0)$.
Answer: $y - y_0 = \left(\frac{dy}{dx}\right)_{(x_0,y_0)} (x - x_0)$.
36.
Write the standard equation of the normal to the curve $y=f(x)$ at the point $(x_0, y_0)$.
Answer: $y - y_0 = \frac{-1}{\left(\frac{dy}{dx}\right)_{(x_0,y_0)}} (x - x_0)$.
37.
If a tangent line to a curve is parallel to the x-axis, its slope $\left(\frac{dy}{dx}\right)$ must be equal to _________________.
Answer: $0$.
38.
If a tangent line to a curve is parallel to the y-axis, the value of $\frac{dx}{dy}$ must be equal to _________________.
Answer: $0$ (or slope $\frac{dy}{dx}$ is infinite/undefined).
39.
Find the slope of the tangent to the line $y = 3x - 5$ at any point.
Answer: $\frac{dy}{dx} = 3$.
40.
Two curves intersect orthogonally if the angle of intersection between their tangents at the point of contact is _________________ degrees.
Answer: $90^\circ$ (or $\frac{\pi}{2}$ radians).
41.
If two curves are orthogonal, the product of their tangent slopes $m_1 m_2$ equals _________________.
Answer: $-1$.
42.
True or False: The normal to a curve is always perpendicular to the tangent at the point of contact.
Answer: True.
43.
The differential of $y$, denoted by $dy$, is defined as $dy = f'(x) \times $ _________________.
Answer: $\Delta x$ (or $dx$).
44.
If $\Delta x$ is a small change in $x$, then the approximate change in $y$ (denoted by $\Delta y$) is calculated as $\Delta y \approx $ _________________.
Answer: $\frac{dy}{dx} \cdot \Delta x$.
45.
The absolute error in a variable $x$ is typically denoted by the symbol _________________.
Answer: $\Delta x$.
46.
The relative error in a variable $x$ is given by the ratio _________________.
Answer: $\frac{\Delta x}{x}$.
47.
The percentage error in a variable $x$ is computed by multiplying the relative error by _________________.
Answer: $100$.
48.
Find the expression for the differential $dy$ for the function $y = x^3$.
Answer: $dy = 3x^2 \, dx$.
49.
Find the expression for the differential $dy$ for the function $y = \sqrt{x}$.
Answer: $dy = \frac{1}{2\sqrt{x}} \, dx$.
50.
If $y = x^2$, calculate the approximate value of $\Delta y$ when $x = 2$ and $\Delta x = 0.01$.
Answer: $\Delta y \approx \frac{dy}{dx}\Delta x = (2x)(0.01) = 2(2)(0.01) = 0.04$.
51.
True or False: For linear functions, $\Delta y$ and $dy$ represent exactly the same numerical value.
Answer: True.
52.
Complete the standard approximation formula: $f(x + \Delta x) \approx f(x) + $ _________________.
Answer: $f'(x) \cdot \Delta x$.
53.
If the radius of a circle increases by a small amount $\Delta r$, write the formula for the approximate change in its area $\Delta A$.
Answer: $\Delta A \approx 2\pi r \cdot \Delta r$.
54.
What is the formula for the relative error in measuring the volume $V$ if the absolute error in measurement is $\Delta V$?
Answer: $\frac{\Delta V}{V}$.
55.
An interior point $c$ in the domain of a function where $f'(c) = 0$ is called a _________________ point.
Answer: Critical (or Stationary) point.
56.
By the First Derivative Test, if $f'(x)$ changes sign from positive to negative as $x$ increases through $c$, then $c$ is a point of local _________________.
Answer: Maximum.
57.
By the First Derivative Test, if $f'(x)$ changes sign from negative to positive as $x$ increases through $c$, then $c$ is a point of local _________________.
Answer: Minimum.
58.
If $f'(x)$ does not change sign as $x$ increases through $c$, then $c$ is neither a maxima nor minima. It is called a point of _________________.
Answer: Inflection.
59.
In the Second Derivative Test, if $f'(c) = 0$ and $f''(c) < 0$, then the point $c$ is a point of local _________________.
Answer: Maximum.
60.
In the Second Derivative Test, if $f'(c) = 0$ and $f''(c) > 0$, then the point $c$ is a point of local _________________.
Answer: Minimum.
61.
If $f'(c) = 0$ and $f''(c) = 0$, we say that the Second Derivative Test _________________ (fails/succeeds).
Answer: Fails (and we must go back to the First Derivative Test).
62.
Find the critical point for the quadratic function $f(x) = x^2 - 4x + 5$.
Answer: $f'(x) = 2x - 4 = 0 \Rightarrow x = 2$.
63.
What is the value of the second derivative $f''(x)$ for the function $f(x) = x^2 - 4x + 5$?
Answer: $f''(x) = 2$.
64.
Based on the previous two questions, does $f(x) = x^2 - 4x + 5$ have a local maximum or a local minimum at its critical point?
Answer: Local Minimum (because $f''(2) = 2 > 0$).
65.
Find the critical point for the function $f(x) = -x^2 + 6x$.
Answer: $f'(x) = -2x + 6 = 0 \Rightarrow x = 3$.
66.
Does the function $f(x) = -x^2 + 6x$ have a local maximum or local minimum at its critical point?
Answer: Local Maximum (because $f''(x) = -2 < 0$).
67.
To find the absolute (global) maximum or minimum of a continuous function in a closed interval $[a, b]$, we must evaluate the function at all critical points and at the _________________ of the interval.
Answer: Endpoints (i.e., at $x = a$ and $x = b$).
68.
True or False: A local maximum value of a function is always the absolute maximum value of the function in its entire domain.
Answer: False.
69.
True or False: The absolute maximum and absolute minimum of a continuous function on a closed interval always exist.
Answer: True (Extreme Value Theorem).
70.
What is the value of the first derivative $f'(x)$ precisely at a stationary point?
Answer: $0$.
71.
If two numbers $x$ and $y$ sum to $10$, write $y$ in terms of $x$.
Answer: $y = 10 - x$.
72.
Using the previous question, write their product $P = xy$ as a function of the single variable $x$.
Answer: $P(x) = x(10 - x) = 10x - x^2$.
73.
If the perimeter of a rectangle is fixed at $20\text{ cm}$, what is the sum of its length $x$ and width $y$?
Answer: $2(x+y) = 20 \Rightarrow x+y = 10$.
74.
Write the area $A$ of the rectangle in Q73 as a function of its length $x$ only.
Answer: $y = 10-x$, so $A(x) = x(10-x)$.
75.
In optimization problems, the function that needs to be maximized or minimized is called the _________________ function.
Answer: Objective.
76.
Write the formula for the volume $V$ of a rectangular box with a square base of side $x$ and a height $h$.
Answer: $V = x^2 h$.
77.
Write the formula for the total surface area $S$ of a closed cylindrical can with radius $r$ and height $h$.
Answer: $S = 2\pi r h + 2\pi r^2$.
78.
Write the formula for the volume $V$ of a cylindrical can with radius $r$ and height $h$.
Answer: $V = \pi r^2 h$.
79.
If a manufacturer wants to minimize the cost of material used to build a box, they need to formulate an equation to minimize the box's _________________ area.
Answer: Surface.
80.
If a manufacturer wants a container to hold the maximum amount of liquid possible, they need to formulate an equation to maximize the container's _________________.
Answer: Volume.
81.
True or False: In applied optimization problems, we use the given constraints to reduce the objective function down to a single independent variable before differentiating.
Answer: True.
82.
Once an objective function $f(x)$ is formed in terms of a single variable, what is the very first calculus step to find its maximum or minimum?
Answer: Find the first derivative $f'(x)$ and set it equal to zero (to find critical points).
