Calculus Volume 1 · free preview
Defining the Derivative
The derivative at a point is the limit of difference quotients, the slope of the tangent line and the instantaneous rate of change, with left and right forms covering one-sided behavior.
1. The difference quotient limit
The difference quotient limit
2. Derivative as velocity and rate
Derivative as velocity and rate
3. One-sided derivatives and failure
One-sided derivatives and failure
4. Leibniz notation
Leibniz notation
5. Definition of the derivative
Which limit defines the derivative of f at the point a?
6. Derivative of x squared at a point
Using the definition, what is the derivative of f(x) = x^2 at x = 3?
7. Velocity interpretation
The position of a particle is s(t) = t^2 meters at time t seconds. What is its instantaneous velocity at t = 2?
8. Absolute value at zero
Why is f(x) = |x| not differentiable at x = 0?
9. Differentiability implies continuity
Which statement about the relationship between differentiability and continuity is correct?
10. Equivalent limit form
Which limit is equivalent to the derivative f prime of a when the second point is named x instead of a + h?
11. Derivative of x at a point
The function f(x) = 2x has constant slope. Using the definition or the slope read directly, what is f prime of 5? Answer with a number.
12. Units of a derivative
The cost C(q) is in dollars for q units. The derivative dC/dq is marginal cost. What is its unit, one word? Answer in lowercase.
13. Right derivative of the absolute value
The absolute value has constant slope 1 to the right of zero. What is its right-hand derivative at 0? Answer with a number.
14. Leibniz symbol
In Leibniz notation the derivative of y with respect to x is written as a fraction of d symbols. What is the numerator's single letter? Answer with one lowercase letter.
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