Calculus Concepts and Functions Quiz

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Questions and Answers

What is the range of the function defined by f: x → 2x² - 1?

{1, 7, 17} (correct)

{-1, 1, 3, 7, 17}

{-1, 3, 7}

{1, 3, 5}

Which statement is true regarding the continuity of a function at a point c?

All conditions need to be met for continuity. (correct)

The limit exists but f(c) is not defined.

f(c) is defined, but the limit does not exist.

lim_(x→c) f(x) = f(c), but f(c) is not defined.

What does the derivative of f(x) = 5^(x^2) represent?

5^(x^2 - 1)

5^(x^2) * ln(5) * 2x (correct)

5^(x^2) * 2x

2x * ln(5)

Find the fourth derivative of the function y = 3x³ - 2x² + x² + 1.

0 Signup and view all the answers

If y is the total cost of manufacturing x units described by y = (x² + 20)/(4x), what is dy/dx?

(-x² - 20)/(4x²) Signup and view all the answers

What is the limit of (tan x)/(sin x) as x approaches 0?

1 Signup and view all the answers

What is the equation of the tangent to the curve at the point (1, 2) if the slope is 5?

y - 2 = 5(x - 1) Signup and view all the answers

Which of the following statements about inverse functions is true?

A function has an inverse if it is bijective. Signup and view all the answers

What can be said about the inverse of a function? A function has an inverse if it is which of the following?

Bijective Signup and view all the answers

In which of the following cases does the limit lim_(x→0) (1 - cos x)/x² converge to a specific value?

When x approaches 0 Signup and view all the answers

What is the result of differentiating the function y = sin(x²) with respect to x?

2x cos(x²) Signup and view all the answers

Which of the following derivatives represents the slope of the curve defined by the equation x²y + xy² + 3x - 13 = 0 at the point (1, 2)?

2 Signup and view all the answers

For which type of function is it true that every element of the domain results in a unique image in the range?

Function Signup and view all the answers

What is the derivative of the function f(x) = cot x?

-csc²x Signup and view all the answers

Which of the following statements regarding the domain of the function f: x → 2x² - 1 is correct given the range is {1, 7, 17}?

Domain can be any real number. Signup and view all the answers

Evaluate the limit lim_(x→-2) (x² - 2x + 3)/(x² + 4x + 4). What result do you obtain?

1 Signup and view all the answers

Which method would be appropriate to find dy/dx for the equation y = (x + y)² + 10?

Implicit differentiation Signup and view all the answers

Determine the result of evaluating the integral ∫sin³x dx.

(3/4) sin x - (1/4) sin(3x) + C Signup and view all the answers

Study Notes

General Instructions

All rough work should be done in the answer booklet only
Answer all questions
Do not write anything except your registration number on this paper.

Question 1

All relations are functions but not all functions are relations
A function has an inverse if it is bijective
Every element of the domain has an image in the codomain

Question 2

Find the domain of a function given its range.
The domain is the set of possible x-values for the inputs
The range is the set of possible y-values for the outputs.

Questions 3

Find the composite functions g[f(x)] and f[g(x)] with given f(x) and g(x)

Question 4

Evaluate the limit of tanx/sinx

Question 5

Find the inverse of the function h(x) = (2x-1)/(3x-1)

Question 6

Use implicit differentiation to find dy/dx for y = (x +y)² + 10.

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Description

Test your understanding of key calculus concepts, including functions, limits, and derivatives. This quiz covers a range of topics such as finding domains, evaluating limits, and using implicit differentiation. Challenge yourself and reinforce your calculus skills.