Calculus: Derivative of f/g(x)
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Questions and Answers

What is the main objective of the Calculus course?

  • To train students to work as mathematicians
  • To provide students with the fundamental concepts of calculus and problem-solving skills (correct)
  • To provide students with a basic understanding of mathematics
  • To teach students how to solve engineering problems

    • What percentage of the total assessment is the continuous assessment?

  • 50%
  • 30%
  • 40% (correct)
  • 20%

    • What is one of the skills students should have after completing the Calculus course?

  • Ability to apply differentiation to solve maxima and minima problems (correct)
  • Ability to solve physics problems
  • Ability to teach calculus
  • Ability to write calculus software

    • What is the weightage of the end of semester examination?

    <p>60%</p> Signup and view all the answers

    What type of assignments will be given to students?

    <p>Both group and individual assignments</p> Signup and view all the answers

    What is one of the topics that will be covered in the Calculus course?

    <p>Differential equations</p> Signup and view all the answers

    What percentage of the continuous assessment is attendance?

    <p>10%</p> Signup and view all the answers

    What is the derivative of sin x?

    <p>cos x</p> Signup and view all the answers

    What is the derivative of cos ax?

    <p>-asin ax</p> Signup and view all the answers

    Who compiled the lecture note?

    <p>Kofi Agyarko</p> Signup and view all the answers

    What is the principal value of arcsin x?

    <p>between -π/2 and π/2</p> Signup and view all the answers

    What is the derivative of tan x?

    <p>sec^2 x</p> Signup and view all the answers

    What is the derivative of csc x?

    <p>-csc x cot x</p> Signup and view all the answers

    What is the derivative of sec x?

    <p>sec x tan x</p> Signup and view all the answers

    What is the concept of velocity in the context of derivatives?

    <p>The rate of change of displacement with respect to time</p> Signup and view all the answers

    What is the definition of instantaneous velocity?

    <p>The rate of change of displacement with respect to time</p> Signup and view all the answers

    What is the derivative of a function f(x) in geometrical terms?

    <p>The slope of the line tangent to the graph of f(x)</p> Signup and view all the answers

    What is the concept of the derivative in physical terms?

    <p>The rate of change of displacement with respect to time</p> Signup and view all the answers

    What is the formula for the instantaneous velocity?

    <p>Instantaneous velocity = lim Δs / Δt as Δt approaches zero</p> Signup and view all the answers

    What is the meaning of the derivative in geometrical terms?

    <p>The slope of the line tangent to the graph of f(x)</p> Signup and view all the answers

    What is the definition of the derivative of a function f(x)?

    <p>The slope of the line tangent to the graph of f(x)</p> Signup and view all the answers

    What is the concept of the derivative in mathematical terms?

    <p>The limit of the secant line as the distance approaches zero</p> Signup and view all the answers

    What is the derivative of tan x?

    <p>sec²x</p> Signup and view all the answers

    What is the integral of sin x cos x?

    <p>Sin²x + c</p> Signup and view all the answers

    What is the integral of tan x sec²x?

    <p>Tan²x + c</p> Signup and view all the answers

    What is the integral of ln x/x?

    <p>(ln x)² + c</p> Signup and view all the answers

    What is the derivative of sec x?

    <p>sec x tan x</p> Signup and view all the answers

    What is the integral of √(1-x²)?

    <p>sin⁻¹ x + c</p> Signup and view all the answers

    What is the integral of sin x?

    <p>-cos x + c</p> Signup and view all the answers

    What is the integral of tan x sec²x?

    <p>tan²x + c</p> Signup and view all the answers

    What is the derivative of 𝑓⁄𝑔 with respect to 𝑥?

    <p>$\frac{𝑔(𝑥) 𝑓'(𝑥) - 𝑓(𝑥) 𝑔'(𝑥)}{𝑔(𝑥)^2}$</p> Signup and view all the answers

    What is the definition of the derivative of ℎ(𝑥) with respect to 𝑥?

    <p>ℎ'(𝑥) = lim (ℎ(𝑥 + ∆𝑥) - ℎ(𝑥))/∆𝑥 as ∆𝑥 → 0</p> Signup and view all the answers

    What is the special case of the derivative of 𝑓⁄𝑔 with respect to 𝑥 when 𝑓(𝑥) = 1?

    <p>$\frac{-𝑔'(𝑥)}{𝑔(𝑥)}$</p> Signup and view all the answers

    What is the formula for the derivative of 𝑓⁄𝑔 with respect to 𝑥 in terms of the derivatives of 𝑓 and 𝑔?

    <p>$\frac{𝑔(𝑥) 𝑓'(𝑥) - 𝑓(𝑥) 𝑔'(𝑥)}{𝑔(𝑥)^2}$</p> Signup and view all the answers

    What is the purpose of the limit as ∆𝑥 → 0 in the definition of the derivative?

    <p>To make the change in 𝑥 infinitesimally small</p> Signup and view all the answers

    What is the chain rule used for?

    <p>To find the derivative of a composite function</p> Signup and view all the answers

    What is the derivative of ℎ(𝑥) = 𝑓(𝑥)⁄𝑔(𝑥) with respect to 𝑥?

    <p>$\frac{𝑔(𝑥) 𝑓'(𝑥) - 𝑓(𝑥) 𝑔'(𝑥)}{𝑔(𝑥)^2}$</p> Signup and view all the answers

    What is the condition for 𝑓(𝑥) and 𝑔(𝑥) to be differentiable?

    <p>They must be both differentiable</p> Signup and view all the answers

    What is the formula for the derivative of 1⁄𝑔(𝑥) with respect to 𝑥?

    <p>$-\frac{𝑔'(𝑥)}{𝑔(𝑥)}$</p> Signup and view all the answers

    What is the purpose of the example in the text?

    <p>To illustrate the formula for the derivative of 𝑓⁄𝑔</p> Signup and view all the answers

    Study Notes

    Derivative of a Function

    • The derivative of a function 𝑓(𝑥) is the limit of the ratio of the change in the function value to the change in 𝑥, as the change in 𝑥 approaches zero.
    • The derivative can be represented as 𝑓′(𝑥) or 𝑑𝑓/𝑑𝑥.
    • The derivative of a function can be used to find the rate of change of the function with respect to 𝑥.

    Differentiation Rules

    • The derivative of a product of functions is the product of their derivatives.
    • The derivative of a quotient of functions is the quotient of their derivatives.
    • The derivative of a function with a constant coefficient is the coefficient times the derivative of the function.

    Geometrical Concept of Derivatives

    • The derivative of a function can be represented as the slope of the line tangent to the graph of the function.
    • The derivative of a function can be used to find the rate of change of the function with respect to 𝑥.

    Physical Concept of Derivatives

    • The derivative of a function can be used to find the rate of change of the function with respect to time.
    • The derivative of a function can be used to find the velocity of an object.

    Derivative of Trigonometric Functions

    • The derivative of sin(𝑥) is cos(𝑥).
    • The derivative of cos(𝑥) is -sin(𝑥).
    • The derivative of tan(𝑥) is sec^2(𝑥).
    • The derivative of cot(𝑥) is -csc^2(𝑥).
    • The derivative of sec(𝑥) is sec(𝑥)tan(𝑥).
    • The derivative of csc(𝑥) is -csc(𝑥)cot(𝑥).

    Differentiation of Inverse Trigonometric Functions

    • The derivative of arcsin(𝑥) is 1/√(1-𝑥^2).
    • The derivative of arccos(𝑥) is -1/√(1-𝑥^2).
    • The derivative of arctan(𝑥) is 1/(1+𝑥^2).
    • The derivative of arccot(𝑥) is -1/(1+𝑥^2).
    • The derivative of arcsec(𝑥) is 1/|𝑥|√(𝑥^2-1).
    • The derivative of arccsc(𝑥) is -1/|𝑥|√(𝑥^2-1).

    Chain Rule

    • The derivative of a composite function is the derivative of the outer function times the derivative of the inner function.

    Course Objectives

    • Find the derivatives of functions from first principle and using the rules of differentiation.
    • Apply differentiation to solve maxima and minima problems.
    • Apply differentiation to solve rates of change problems.
    • Find the derivatives of logarithmic functions.
    • Evaluate integrals using the fundamental theorem of calculus.
    • Evaluate integrals using advanced techniques of integration.
    • Find the partial derivatives of functions of several variables.
    • Solve basic problems in ordinary differential equations.
    • Obtain the Laplace transform of simple standard expressions.
    • Use the first shift theorem to find the Laplace transform of simple expressions multiplied by an exponent.
    • Identify and solve problems involving Gamma and Beta functions.

    Assessment

    • Continuous assessment (40%) and end-of-semester examination (60%).
    • Continuous assessment includes attendance, group assignments, and individual assignments.

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    Find the derivative of f/g(x) using limits and prove its differentiability. Practice your calculus skills with this challenging quiz!

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