Calculus Notes on Derivatives and Applications
Derivatives and Related Concepts
Section 1: Basic Derivatives Calculations
Problems: Calculate y'. (Problems 1 to 54)
Problem 1:
Function: $y = (x^2 + x^3)^4$
Derivative: $y' = ?$Problem 2:
Function: $y = x^2 - x + 2$
Derivative: $y' = ?$Problem 3:
Function: $y = rac{1}{ ext{√}x}$
Derivative: $y' = ?$Problem 4:
Function: $y = an x$
Derivative: $y' = ext{sec}^2(x)$Problem 5:
Function: $y = x^2 ext{sin}(x)$
Derivative: $y' = 2x ext{sin}(x) + x^2 ext{cos}(x)$Problem 6:
Function: $y = x ext{cos}^{-1}(x)$
Derivative: $y' = ext{cos}^{-1}(x) + rac{-x}{ ext{√}(1-x^2)}$Problem 7:
Function: $y = rac{ an x}{1 + ext{cos} x}$
Derivative: $y' = ?$Problem 8:
Function: $x e^y = y ext{sin}(x)$
Implicit Derivative: Solve for $y'$Problem 55:
Function: If $f(t) = ext{√}(4t + 1)$, calculate $f''(2)$.
Section 2: Tangent Lines and Curve Analysis
Problems: Find equations of tangent lines to the curves at specific points. (Problems 60 to 63)
Problem 61:
Function: $y = 4 ext{sin}^2(x)$ at $(1)$Problem 62:
Function: $y = rac{x^2 - 1}{x^2 + 1}$ at $(0, -1)$Problem 63:
Function: $y = ext{√}(1 + 4 ext{sin}(x))$ at $(0, 1)$
Section 3: Finding Higher Order Derivatives
Problems: Calculate higher order derivatives. (Problems 66 to 70)
Problem 66:
Function: $f(x) = x e^{ ext{sin}(x)}$, find $f'(x)$.Problem 67:
Function: $f(x) = x ext{√}(5 - x)$
a. Find $f'(x)$.
b. Find the tangent lines at the points $(1, 2)$ and $(4, 4)$.Problem 68:
Function: $f(x) = 4x - an x$, find $f'(x)$ and $f''(x)$.
Section 4: Applications of Derivatives
Problems: Use derivatives in optimization and motion analysis (Problems 91 to 94).
Problem 91:
Function: $s = A e^{ct} ext{cos}(wt + ext{θ})$.Find the velocity and acceleration of the object.
Problem 92:
Function: $x(t) = ext{√}(b^2 + c^2 t^2)$, where $b$ and $c$ are constants.
a. Find velocity and acceleration.
b. Show that the particle always moves positively.
Section 5: Understanding Rates of Change
Problems: Conceptual understanding and application of rates of change. (Problems 101 to 104).
Problem 101:
Function: Volume of a cube increasing at $10 ext{cm}^3/ ext{min}$.Find how fast the surface area is increasing at the edge of $30 ext{cm}$.
Problem 102:
Function: A paper cup shaped like a cone; water poured in at $2 ext{cm}^3/ ext{s}$, find how fast the water level rises when 5 cm deep.Problem 103:
Scenario: A balloon and a boy cycling, calculate the distance increasing after 3 seconds.Problem 104:
Scenario: Waterskier leaving a ramp, find the rate of ascent.
The problems in these notes are designed to cover various aspects of differential calculus, from basic calculations to advanced applications:
Section 1: Basic Derivatives Calculations
This section asks you to find the first derivative (y' or f'(x)) of different functions using various differentiation rules (e.g., power rule, chain rule, product rule, quotient rule, trigonometric derivatives, inverse trigonometric derivatives).
For Problem 1, 2, 3, and 7, you need to apply the appropriate differentiation rules to find y'.
Problem 8 specifically requires you to use implicit differentiation to solve for y'.
Problem 55 asks for the second derivative (f''(2)) of a function, which means you'll differentiate the function twice and then evaluate it at t=2 (noting the variable t for time).
Section 2: Tangent Lines and Curve Analysis
This section focuses on finding the equation of a tangent line to a given curve at a specified point.
To solve these problems, you typically need to:
Find the first derivative of the function (y' or slope m).
Evaluate the derivative at the given x-coordinate to find the slope of the tangent line at that point.
Use the point-slope form y - y1 = m(x - x1) to write the equation of the tangent line.
Problem 61 provides only the x-coordinate, so you'd also need to find the corresponding y-coordinate using the original function.
Section 3: Finding Higher Order Derivatives
This section extends the concept of differentiation to finding derivatives beyond the first order.
Problem 66, 67a, and 68 ask for the first derivative (f'(x)).
Problem 67b combines finding derivatives with Section 2's concept, asking for tangent lines at specific points.
Problem 68 specifically asks for both the first (f'(x)) and second (f''(x)) derivatives of the function.
Section 4: Applications of Derivatives
This section applies derivatives to real-world scenarios, particularly in motion analysis.
Problem 91 and 92a involve a position function (s or x) and ask for the velocity (first derivative of position with respect to time) and acceleration (second derivative of position with respect to time).
Problem 92b requires you to analyze the sign of the velocity to show the particle's direction of movement.
Section 5: Understanding Rates of Change
This section deals with related rates problems, where you are given the rate of change of one quantity and need to find the rate of change of another quantity that is related to the first.
These problems typically require you to:
Identify the given rates and the rate you need to find.
Establish a relationship (an equation) between the changing quantities.
Differentiate the equation with respect to time using the chain rule.
Substitute the known values and solve for the unknown rate.
Problem 101, 102, 103, and 104 all involve scenarios where quantities are changing over time, and you're asked to find how fast a specific quantity is changing at a given instant.