Calculus for Life Sciences Quiz
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Questions and Answers

What percentage of the total course assessment is allocated to the final exam?

  • 20%
  • 10%
  • 30%
  • 40% (correct)

    • Karim Kamal is the teaching assistant for the course.

    False

    What is the primary role of mathematics in Pharmaceutical Engineering?

    To describe physiological processes and develop mathematical models.

    The email address for the teaching assistant is ______.

    <p><a href="mailto:[email protected]">[email protected]</a></p> Signup and view all the answers

    Match the following course components with their corresponding percentages:

    <p>Assignments = 10% Quizzes = 20% Midterm = 30% Final Exam = 40%</p> Signup and view all the answers

    Which textbook focuses on Calculus for Life Sciences?

    <p>Calculus for the Life Sciences</p> Signup and view all the answers

    The course includes topics such as Complex Numbers and Statistics.

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

    What is the office location of Dr. Karim Kamal?

    <p>S1.304</p> Signup and view all the answers

    What is the derivative of $y = tan(x)$?

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

    The derivative of $y = x^2 sin(x)$ is $2x sin(x) + x^2 cos(x)$.

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

    What is the derivative of the function $y = x + tan(x)$?

    <p>1 + sec^2(x)</p> Signup and view all the answers

    The derivative of $y = \frac{sin(x)}{1 + cos(x)}$ is _____

    <p>$\frac{cos(x)(1 + cos(x)) - sin(x)(-sin(x))}{(1 + cos(x))^2}$</p> Signup and view all the answers

    Match the following derivatives with their functions:

    <p>sec^2(x) = tan(x) cos(x) = sin(x) 2x = x^2 1/(1 + cos(x))^2 = sin(x)/(1 + cos(x))</p> Signup and view all the answers

    Which of the following correctly represents the rule used for finding the derivative of a product of functions?

    <p>u'v + uv'</p> Signup and view all the answers

    The derivative of $y = x + tan(x)$ is simply $sec^2(x)$.

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

    What is the derivative of $y = x^2 sin(x)$?

    <p>$2x sin(x) + x^2 cos(x)$</p> Signup and view all the answers

    What is the solution to the equation $\frac{x + 5}{3} = -27$?

    <p>$-8$</p> Signup and view all the answers

    An interval can only be bounded.

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

    What do the symbols <, >, ≤, and ≥ represent?

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

    The solution of an inequality is the set of all values that make the inequality a true statement, typically represented as an __________.

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

    Match the types of intervals with their descriptions:

    <p>Bounded = Corresponds to a line segment on the number line Unbounded = Corresponds to a ray on the number line Closed interval = Includes endpoints Open interval = Does not include endpoints</p> Signup and view all the answers

    Which of the following is an example of an unbounded interval?

    <p>]1, ∞[</p> Signup and view all the answers

    When multiplying or dividing both sides of an inequality by a negative number, the inequality sign stays the same.

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

    What is an example of a bounded interval?

    <p>]3, 7[</p> Signup and view all the answers

    What is the first step in differentiating the function $y = x^{3} (x + 3)^{3} (3x - 1)^{5}$?

    <p>Take the natural logarithm of both sides.</p> Signup and view all the answers

    The derivative of a function can be found by directly applying the product and chain rule without any transformations.

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

    What is the product rule formula used for finding the derivative of two functions?

    <p>If $u$ and $v$ are functions, then $(uv)' = u'v + uv'$</p> Signup and view all the answers

    The derivative $\frac{dy}{dx}$ can be expressed as a function of $y$ and its components, specifically $\frac{1}{y} \frac{dy}{dx} = \frac{1}{2x} + \frac{1}{3(x + 3)} + \frac{1}{5(3x - 1)}$. Fill in the blank: To isolate $\frac{dy}{dx}$, we multiply both sides by _____ .

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

    Match each operation with its corresponding result in differentiation.

    <p>Power rule = If $f(x) = x^n$, then $f'(x) = n x^{n-1}$ Product rule = If $u$ and $v$ are functions, $(uv)' = u'v + uv'$ Chain rule = If $f(g(x))$, then $f'(g(x))g'(x)$ Quotient rule = If $u$ and $v$ are functions, $(\frac{u}{v})' = \frac{u'v - uv'}{v^2}$</p> Signup and view all the answers

    What is the value of $a^0$ if $a eq 0$?

    <p>1</p> Signup and view all the answers

    $0^3 = 0$ is a correct statement.

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

    What is the result of $(-10)^2$?

    <p>100</p> Signup and view all the answers

    For any real number $a$, $a^{-n} = \frac{1}{a^{______}}$, where $a \neq 0$.

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

    Which of the following is a valid law of exponents?

    <p>$a^m a^n = a^{m+n}$</p> Signup and view all the answers

    Match the expression with their simplified forms:

    <p>$x^3 / x^{-2}$ = $x^5 $x^{4} y^{-1}$ = $\frac{x^4}{y} $3^2 \cdot 3^4$ = $3^6 $5^{-1}$ = $\frac{1}{5}</p> Signup and view all the answers

    What is the simplified result of $x^{-1}$?

    <p>\frac{1}{x}</p> Signup and view all the answers

    The expression $(1,000,000)^0$ equals ______.

    <p>1</p> Signup and view all the answers

    What is the second derivative of the function $f(x) = 2x^3 - 15x^2 + 24x - 7$?

    <p>12x - 30</p> Signup and view all the answers

    L'Hopital's Rule can be applied to all limits regardless of the forms 0/0 or ∞/∞.

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

    At the critical point $x = 1$, what type of extremum does the function $f(x)$ have?

    <p>local maximum</p> Signup and view all the answers

    The limit resulting from the indeterminate form 0/0 can often be resolved using __________.

    <p>L'Hopital's Rule</p> Signup and view all the answers

    Match the function to its corresponding limit type:

    <p>$lim_{x o 2} \frac{x^2 - 4}{x - 2}$ = Indeterminate form 0/0 $lim_{x o 0} \frac{1 - ext{sin} x}{x^2}$ = Indeterminate form 0/0 $lim_{x o rac{ ext{π}}{2}} \frac{ ext{cos} x}{ ext{tan} x}$ = Indeterminate form 0/0 $lim_{x o 0} \frac{x^2}{x - 4/3}$ = Not an indeterminate form</p> Signup and view all the answers

    What happens to $f''(x)$ at the critical point $x = 4$?

    <p>It is greater than 0</p> Signup and view all the answers

    Indeterminate forms include 0 x ∞ and ∞ - ∞.

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

    Determine $f(4)$ for the function $f(x) = 2x^3 - 15x^2 + 24x - 7$.

    <p>-23</p> Signup and view all the answers

    Study Notes

    In-Class Assignment (2)

    • Due date: The week starting 7/12/2024
    • Location: Tutorials
    • Content: Lectures 8, 9, and Worksheet 6

    Lecture 10 – Outline

    • Chain Rule
    • Differentiation of Trigonometric and Hyperbolic Functions
    • Chain Rule in Trigonometric and Hyperbolic Functions

    Rules of Differentiation

    • Rule 9 Chain Rule
    • If there are two differentiable functions f(x) and g(x):
      • If F(x) = (f°g)(x), then the derivative of F(x) is F'(x) = f'(g(x))∙g'(x)
    • If y = f(u) and u = g(x), then the derivative of y is dy/dx = (dy/du) ∙ (du/dx).

    Exercise 3

    • Find y' for given functions:
        1. y = (x² + 4x)³
        1. y = (5x⁴ + 2x² + 5)⁵
        1. y = (e³x + 3x - 1)²
        1. y = √x³ + 5x
        1. y = 6 √e²x

    Exercise 3 - Sol.

    • Solutions to y' calculation for the given functions in exercise 3.

    Exercise 4

    • Find dy/dx in the given functions
        1. y = ex³ ln(2x + 1)
        1. y = e√x + 2x² + 7x
        1. y = (1 + √x + 1)¹⁰

    Exercise 4 – Sol.

    • Solutions to dy/dx calculation for the given functions in exercise 4.

    Exercise 5

    • Find dy/dx in the following functions:
        1. y = ln(x² - 6)
        1. y = 1 + [ln(x)]²
        1. y = log₂ √x + 2
        1. y = log x/(x+1)

    Exercise 5 – Sol.

    • Solutions to dy/dx calculation for the given functions in exercise 5.

    Exercise 5.1 - Sol.

    • Solutions to dy/dx calculation for the given function in exercise 5.1

    Derivatives of Trigonometric Functions

    • d/dx (sinx) = cosx
    • d/dx (cosx) = -sinx
    • d/dx (tanx) = sec² x
    • d/dx (secx) = secx tanx
    • d/dx (cotx) = -csc² x
    • d/dx(cscx) = -cscx cotx

    Example

    • Prove that d/dx (tanx) = sec² x.

    Example – Sol.

    • Solution to the proof of d/dx (tanx) = sec² x.

    Exercise

    • Find y' if:
        1. y = √x + tanx
        1. y = x² sinx
        1. y = sinx/(1 + cosx)

    Exercise - Sol.

    • Solutions to y' calculation for the given functions in exercise.

    Derivatives of Trig./Hyperbolic Functions

    • d/dx (sinx) = cosx
    • d/dx(sinh x) = cosh x
    • d/dx (cosx) = -sinx
    • d/dx (cosh x) = sinh x
    • d/dx (tan x) = sec² x
    • d/dx (tanh x) = sech² x
    • d/dx (cot x) = -csc² x
    • d/dx (coth x) = -csch² x
    • d/dx (sec x) = sec x tan x
    • d/dx (sech x) = -sech x tanh x
    • d/dx (csc x) = -csc x cot x
    • d/dx (csch x) = -csch x coth x

    Example

    • Prove that d/dx(sinh x) = cosh x.

    Example – Sol.

    • Solution to the proof of d/dx(sinh x) = cosh x.

    Exercise

    • Find y' if:
      1. y = √x + tanx
      1. y = x² sin x
      1. y = sinx/(1 + cos x)

    Exercise - Sol.

    • Solutions to y' calculation for each of exercise's given functions.

    Summary: Using the Chain Rule

    • Summary relating differentiation chain rule with trig and hyperbolic functions.

    Exercise

    • Find dy/dx for:
        1. y = 4 cos x³
        1. y = √x³ + csc x
        1. y = sin(√1 + cos2x)
        1. y = tan(4x³ + x)

    Exercise - Sol.

    • Solutions to dy/dx calculation for given functions.

    Quiz (2)

    • Date: Sunday 22/12/2024
    • Time: Second Slot (e.g., 1:30 PM - 3:00 PM)
    • Topics: Lectures 10, 11, 12, Worksheets 7 & 8

    Lecture 11 - Outline

    • Applications of differentiation
      • Implicit Differentiation
      • Logarithmic Differentiation
    • Relative extrema and critical points of a function
    • First derivative test

    Implicit Differentiation

    • If y = f(x) , y is expressed explicitly as a function of x.
    • In some cases x and y are related to each other through an equation where it's difficult to express y as an explicit function of x.
    • For example x² + xy +y² = 1.
    • Three steps to implicitly Differentiation:
        1. Differentiate both sides of the equation with respect to x, treating y as a differentiable function of x.
        1. Collect the terms with dy/dx on one side of the equation.
        1. Solve for dy/dx.

    Example (1)

    • Find y' = dy/dx if: x³ + y³ - 9xy = 0
    • Find y' = dy/dx if: y + ln(xy) = 1

    Example (2)

    • Determine the equation of the tangent and normal lines at the point (2, 4) to the folium of Descartes given by x³ +y³
    • 9xy = 0.

    Example (3)

    • Find y' if x³ + y³ = 6xy. Hence, find the equation of the tangent line to the curve at the point (3,3).

    Logarithmic Differentiation: Case 1

    • If we have a complicated expression with multiple products/quotients.
    • Three steps for Logarithmic Differentiation:
        1. Take the natural logarithm of both sides.
        1. Simplify the resulting expression (using the rules).
        1. Differentiate implicitly.

    Example

    • Find f'(x) if:
    • 3 ( √x² - 8. √x³ + 1 ) ( √x² - 8 / ( ( x⁶ -7x +1 ))) .

    Example - Sol.

    Exercise 1

    • Find y' for the following function:
 y = √x √x+3√(3x−1)/
    • Given the following functions:
      1. y = ln(x² – 6)
      1. y=1 + [ln(x)]²
      1. y= log₂ √x + 2
      1. y = log₂(x/(x + 1))
      1. y = log10(1/√x)

    Exercise 1 - Solutions

    Lecture 12 – Outline

    • Relative extrema and critical points of a function
    • Second derivative test
    • L'Hopital's Rule

    The Second Derivative Test for Local Extrema

    • Suppose f'' is continuous on an open interval containing x = c.
      1. If f'(c) = 0 and f''(c) < 0, then f has a local maximum at x = c.
      1. If f'(c) = 0 and f''(c) > 0, then f has a local minimum at x = c.
      1. If f'(c) = 0 and f''(c) = 0, then the test fails.

    Example 2

    • Find the points of local maximum and minimum values for the function f(x) = 2x³ - 3x² - 12x, using the second derivative test

    Example 2 - Sol.

    Exercise 2

    • The function f(x) = 2x³ - 15x² + 24x - 7 has critical points x = 1 and x = 4.  Use the second derivative test for extrema to determine whether f(1) and f(4) are maxima, minima or the test fails.

    Exercise 2 - Sol.

    Indeterminate Forms and L'Hôpital's Rule

    • Limits that result in the forms 0/0 or ∞/∞.
    • Use L'Hôpital's rule to find the limit.

    Examples for the Standard Case of 0/0

    Examples for the Standard Case of ∞/∞

    Exercise 3

    • Evaluate the following limits:
        1. lim X/ex, x→∞
        1. lim (1 − ln x)/ex, x→0+

    Exercises

    • Evaluate each of the following limits:
        1. lim x² – 9 / (x – 3)(x + 3) , x→3
        1. lim x² + 5x + 4 / (x − 1)(x + 1)(x - 5) x→1
        1. lim √x + 4 − 2 / x , x→0.

    Limit at Infinity

    • The limit of a rational function as x → +∞ and as x → -∞.

    Exercises

    • Evaluate the following limits:
        1. lim (4x² - x²)/ (3x³ - 5) , x→∞
        1. lim (−x² − x + 1) / (x + 1) , x→∞

    An Interesting Limit

    • Limit of sin θ / θ as θ → 0.

    Exercise 3

    • Find the following limits:
        1. lim sin (2x) / sin(3x), x→0
        1. lim tan(2x)/sin(3x), x→0

    Application: Internet Connectivity

    • The percentage of US households connected to the internet can be modeled by the given formulas for N(t) and E(t). Estimate the given limits and interpret the results.

    Application: Foreign Trade

    • Annual US imports from China and US exports to China in the same years can be approximated by the given formulas for I(t) and E(t). Numerically estimate the given limits and interpret the results.

    Lecture 9 – Outline

    • Differentiation definition
    • The geometric meaning of the derivative.
    • Differentiation rules 
      • Derivative of a constant function
      • Power rule
      • Constant multiple rule
      • Product rule
      • Quotient rule

    Differentiation

    • The derivative is used to find the slope of a function at a point.
    • The derivative is used to find the instantaneous rate of change of a function at a point.
    • The derivative is computed by finding the limit of the difference quotient as Δx approaches 0.

    Differentiation

    • f'(x), y', dy/dx, df/dx, all represent the derivative of a function with respect to x.

    Rule 1 – Derivative of a Constant Function

    • If f(x) = c, then f'(x) = 0.
    • Example: d(3)/dx = 0.   d(-1)/dx = 0.

    Rule 2 – Power Rule for Positive Integers

    • If n is a positive integer and f(x) = xⁿ, then d(xⁿ)/dx = nxⁿ⁻¹.
    • Examples:  d(x⁵)/dx = 5x⁴, d(x²)/dx = 2x, d(x¹⁵)/dx = 15x¹⁴, d(x²⁰²⁰)/dx = 2020x²⁰¹⁹.

    Rule 2 – Power Rule for Any Real Number n

    • If n is any real number, then d(xⁿ)/dx = nxⁿ⁻¹

    • Examples: f(x) = x⁻³, then f'(x) = -3x⁻⁴= -3/ x⁴, f(x) = √x then f'(x) = 1 / (2√x), f(x) = x³/√x then f'(x) = x⁵/3.

    Rule 3 – Constant Multiple Rule

    • If g(x) is a differentiable function of x, c is a constant, and f(x) = c g(x), then d(cg(x))/dx = c d(g(x))/dx.
    • Examples: d(7x²)/dx = 14x, d(3x⁵)/dx = 15x⁴, d(7x⁷)/dx = 49x⁶

    Exercise 4

    • Find f'(x), if:
        1. f(x) = 5x³
        1. f(x) = 3√x
        1. f(x) = 1/(2√x²)

    Exercise 5

    • Find the derivative of the following functions, with respect to x:
        1. f(x) = x³/30 − x⁸⁰/80+ 99.
        1. f(x) = 4x⁶-3x⁵−10x² + 5x+16.

    Exercise 6

    • Find the points at which the tangent lines to the curve y = x⁴ − 2x² + 2 are horizontal.

    Exercise 7

    • Find the tangent line to the curve y = (1)/x at the point x = 2.

    Rule 4-Derivative Sum Rule

    • If u and v are differentiable functions of x and f(x) = u(x) + v(x), then d(u + v) / dx = d(u) /dx + d(v)/dx
    • Examples: d(5x² + 7x − 6)/dx = 10x + 7, d(x² + 12x)/dx = 4x³ + 12.

    Exercise 8

    • Find the derivative of the following functions with respect to x
        1. f(x) = √x(x² + 3x - 1)
        1. f(x) = 2√x³ (x⁵ − 2)
        1. f(x) = (x⁹ √x⁶)

    Exercise - 9

    • Find the derivative of the following functions with respect to x
      • f(x) = x² - x + 5 / (x³ - 8x+2)

    Rule 6 - Derivative Quotient Rule

    • If u and v are differentiable functions of x, and if v(x) ≠ 0, then the quotient u/v is differentiable at x, and d ( u/v)/ dx = (v.du – u.dv) / v².

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