Calculus Functions and Derivatives Quiz

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Questions and Answers

What is the domain of the function f(x) = log x?

x ≥ 0
x > 0 (correct)
x < 0
All real numbers

Which of the following statements is true regarding the Mean Value Theorem?

It states that there exists a point where the instantaneous rate of change equals the average rate of change. (correct)
It can be applied to non-differentiable functions.
It applies only to continuous functions.
It's applicable only on closed intervals.

Using the Mean Value Theorem, what can be concluded about | sin a − sin b| with respect to |a − b|?

| sin a − sin b| < |a − b|
| sin a − sin b| ≤ |a − b| (correct)
| sin a − sin b| > |a − b|
| sin a − sin b| = |a − b|

Flashcards

Domain of a function

The set of all possible input values (x-values) for which a function is defined.

Composite function (f ◦ g)(x)

A function formed by applying one function (g) to the input, and then applying a second function (f) to the result.

Mean Value Theorem

If a function is continuous on a closed interval and differentiable on the open interval, there exists a point within the interval where the instantaneous rate of change equals the average rate of change over the entire interval.

Implicit Differentiation

A technique for finding the derivative of a function that is defined implicitly (e.g., in an equation) rather than explicitly.

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Differentiation of trigonometric functions

The process of finding the rate of change of trigonometric functions with respect to an input variable (such as x).

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Study Notes

Functions and their Domains

f(x) = log x, g(x) = 1/x
Domain of f(x): x > 0
Domain of g(x): x ≠ 0
(f°g)(x) = f(g(x)) = log(1/x) = -log(x)

Composite Functions

(f°g)(-1) is undefined as x = -1 is outside the domain of g(x)

Mean Value Theorem

For a differentiable function f(x), there exists a c such that f'(c) = (f(b) - f(a))/(b - a) in [a, b]
|sin a - sin b| < |a - b| (proven using Mean Value Theorem)

Differentiable Function

f(0) = 5, -1 < f'(x) < 3
-5 ≤ f(10) (proven by using Mean Value Theorem)

Motion of a Car

Car passes a camera at A with speed 50 km/h
One hour later, the car passes another camera.

Equation of a Circle

x² + y² = 25
Find acceleration at a specific point

Implicit Differentiation

Differentiating x² - y² = 1 implicitly shows that y" = ...

Tangent to a Curve

Find points on the curve y = 2x³ - 3x² - 12x + 20 where the tangent is parallel to the x-axis

Derivatives

Find dy/dx for multiple functions:
- y² = cos(sin√(x² + 1))
- x²/ (x + y) = x - y
- y = cos⁻¹(2x)
- y = x³ ln(2x)
- y = x^x
- y = (x² + 9)²(x - 3)⁴/(x² + 2)

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Description

Test your understanding of functions, their domains, and the Mean Value Theorem in this comprehensive calculus quiz. Explore concepts such as composite functions, implicit differentiation, and the motion of a car, alongside equations of circles and derivatives. Perfect for students looking to reinforce their calculus knowledge.