Graph Interpretation
If the graph of f’(x) is above the x-axis, what is happening to the graph of f(x)?
Find the average rate of change of f(x)=1/x on the interval [1, 4].
-1/4
If you graph f'(x) on a calculator and find that it crosses the x-axis from below to above at x=2, what structural feature does the original graph f(x) have at x = 2?
A local minimum (because f'(x) changes signs from negative to positive)
What does the definition of a normal line tell you about its relationship to a tangent line at the exact same point of tangency?
They are perpendicular (their slopes are opposite reciprocals)
Which tech entrepreneur named his son X Æ A-12?
Elon Musk.
If a function (x) has a sharp corner or a cusp at x=2, what can you say about the graph of its derivative, f'(x), at x=2?
f'(x) does not exist at x=2
Find the instantaneous rate of change of f(x) = 2x^3-5x at x=1.
1
When using a calculator to find the absolute maximum of a continuous function on a closed interval, you locate all the points where the derivative equals zero. What other locations must you conceptually check on the calculator to guarantee you found the absolute maximum?
The endpoints of the interval
If f(x)=sin(x), what is the exact value of its derivative at x=pi?
-1.
According to the unwritten rules of the universe, if a piece of toast falls off the kitchen counter, which side is mathematically guaranteed to hit the floor?
The buttered side.
If the graph of f(x) is linear with a constant positive slope, what does the graph of its derivative f'(x) look like?
A horizontal line located above the x-axis (at y=m).
Find the equation of the tangent line to f(x)=sqrt(x) at x=4.
y=1/4x+1
A student uses their calculator's numerical derivative feature at a specific point on a graph and gets a value of -5,000,000. Conceptually, what is this extreme value indicating about the behavior of the original function's graph at that exact spot?
The graph is decreasing incredibly fast / it is approaching a vertical tangent line (an infinite or undefined slope).
Find the slope of the normal line to the curve f(x)=e^x at the point (0, 1).
-1.
In the cinematic masterpiece Shrek, what asset of nature does Shrek famously compare ogres to because "they have layers"?
Onions
If a function f(x) has a horizontal tangent line at x=3, where does the graph of f'(x) cross or touch?
The graph of f'(x) will have an x-intercept at x=3 (it equals 0).
Use the definition of the derivative to find f'(x) for f(x)=3x+2
3.
If a calculator tells you that a function's average rate of change on [1, 5] is exactly equal to its instantaneous rate of change at x=3, what major Calculus theorem did the calculator just visually demonstrate for you?
The Mean Value Theorem (MVT)
Find the value of c guaranteed by the Mean Value Theorem for f(x)=x^3 on [0,2].
2(sqrt(3))/3 or 1.155
In 2009, what became the first Morse code character to be added since WWII?
The “@” symbol
If the graph of a derivative f'(x) is a horizontal line along the x-axis (y=0), what does the original function f(x) look like?
A horizontal line (f(x) = C, a constant function).
Find d/dx(ln(x)+e^x)
1/x+e^x
If you are using your calculator to graph and find where a tangent line is horizontal, what target value are you searching for on the graph of f'(x)?
The zero/x-intercept of the f'(x) graph.
If f(x) and g(x) are inverse functions, f(2) = 5, and f'(2) =3, what is the value of the derivative of the inverse function at x = 5 (meaning, find g'(5))?
1/3.
The classic cartoon character Donald Duck has a middle name. It is the only major Disney character middle name that represents a literal, structural component of a ship. What is it?
Fauntleroy.