Calculus Volume 1 · free preview
The Derivative as a Function
Treating the derivative pointwise produces a new function f prime whose domain is where the limit exists, whose graph reveals monotonicity and concavity, and whose zeros locate critical points.
1. From number to function
From number to function
2. Differentiability on intervals
Differentiability on intervals
3. Higher derivatives
Higher derivatives
4. The derivative as a standalone object
The derivative as a standalone object
5. The derivative of the derivative
The derivative of f is itself a function. What is the standard name of the derivative of f prime?
6. Reading the sign of f prime
Wherever the derivative f prime is negative on an interval, what must the graph of f be doing there?
7. Where f prime is zero
The derivative f prime of a function equals zero at x = 2. What does that tell us about the graph of f at x = 2?
8. Second derivative of position
In kinematics, position s(t) differentiates to velocity v(t). What does the second derivative s double prime represent?
9. Domain of the derivative
Which statement about the domain of f prime compared to the domain of f is always correct?
10. Sign pattern matching
The derivative f prime is negative on the interval (-2, 1) and positive on (1, 4). What does f do on those intervals?
11. Derivative of x cubed at a point
The function f(x) = x^3 has derivative f prime of x = 3x^2. What is f prime of 2? Answer with a number.
12. Constant slope
A function has f prime of x = 5 for every x. What is the slope of its tangent line at x = 7? Answer with a number.
13. Second derivative of a quadratic
For f(x) = x^2, the first derivative is 2x. What is f double prime of x, its second derivative? Answer like 6x or 0.
14. Order of the jerk
Jerk is the derivative of acceleration, which is itself the second derivative of position. What order derivative of position is jerk, as a number?
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