Calculus for Life Sciences Quiz

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
  • 2. y = (5x⁴ + 2x² + 5)⁵
  • 3. y = (e³x + 3x - 1)²
  • 4. y = √x³ + 5x
  • 5. 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)
  • 2. y = e√x + 2x² + 7x
  • 3. 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)
  • 2. y = 1 + [ln(x)]²
  • 3. y = log₂ √x + 2
  • 4. 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
  • 2. y = x² sinx
  • 3. 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
• 2. y = x² sin x
• 3. 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
  • 2. y = √x³ + csc x
  • 3. y = sin(√1 + cos2x)
  • 4. 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.
  • 2. Collect the terms with dy/dx on one side of the equation.
  • 3. 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.
  • 2. Simplify the resulting expression (using the rules).
  • 3. 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)
• 2. y=1 + [ln(x)]²
• 3. y= log₂ √x + 2
• 4. y = log₂(x/(x + 1))
• 5. 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.
• 2. If f'(c) = 0 and f''(c) > 0, then f has a local minimum at x = c.
• 3. 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→∞
  • 2. lim (1 − ln x)/ex, x→0+

Exercises

• Evaluate each of the following limits:
  • 1. lim x² – 9 / (x – 3)(x + 3) , x→3
  • 2. lim x² + 5x + 4 / (x − 1)(x + 1)(x - 5) x→1
  • 3. 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→∞
  • 2. 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
  • 2. 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
  • 2. f(x) = 3√x
  • 3. 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.
  • 2. 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)
  • 2. f(x) = 2√x³ (x⁵ − 2)
  • 3. 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.