Calc_1

Lab Assignments Overview

Lab Assignment 12

  1. Ladder Problem

    • A 13ft ladder slips against a wall.

    • When the top is 5ft above ground, base slides at 1 ft/s.

    • Find: How fast is the top falling?

  2. Kite Problem

    • Kite at constant height of 120ft.

    • Angle of string decreases at 1/119 radians/sec.

    • Find: Rate of changing distance when 169ft of string is let out.

  3. Wine Aerator Problem

    • Wine poured at 10mL/s, drains at 20mL/s.

    • Aerator has radius of 3cm and height of 5cm.

    • Find: Rate of change of wine depth when depth = 2cm.

  4. Box Volume Problem

    • Square tin (18cm sides) is converted into a box by cutting corners.

    • Objective: Maximize the volume of the box.

  5. Inscribed Rectangle in Triangle

    • Isosceles triangle with base 6cm and height 4cm.

    • Find: Area of largest rectangle inscribed in triangle.

  6. Fencing Cost Minimization

    • Farmer's fencing project for an area of 4320m² divided into 3 equal pens.

    • Outer fencing costs $10/m, inner costs $2/m.

    • Find: Minimum cost of fencing project.

Lab Assignment 8

  1. Curve Analysis

    • Find formula for dy/dx and tangent equations at x = 0.

  2. Power Rule with Logarithmic Differentiation

    • Prove that d/dx (x^n) = nx^(n-1).

  3. Complex Differentiation

    • Differentiate y = x(log x)^x + arctan(x^1/x).

  4. Parametric Curves

    • Find points where tangent lines are perpendicular.

  5. Level Curves

    • For f(x, y) = x^2y + xy^2, find horizontal and vertical tangents at z = 2.

  6. Arc Hyperbolic Functions

    • Prove the formula for d/dx (arccosh(x)).

  7. Parametric Curve Analysis

    • Find slopes of tangent lines at specific points.

Lab Assignment 7

  1. Particle Motion

    • Analyze position s(t), velocity and acceleration at t = 3π/4.

  2. Differentiation Methods

    • Different methods to differentiate specific functions without certain rules.

  3. Tangent Line Analysis

    • Determine tangent lines that pass through the origin for f(x) = x^2 + 4.

Lab Assignment 5

  1. Limit Calculations

    • Calculate lim x→0 tan(3x) sin(7x).

  2. Function Discontinuities

    • Analyze discontinuities and horizontal asymptotes of given functions.

  3. Parametric Curve Discontinuities

    • Identify and classify discontinuities of specified parametric functions.

Lab Assignment 4

  1. Discontinuity Classification

    • Identify and classify discontinuities of given piecewise functions.

  2. Limit Evaluations

    • Evaluate limits as x approaches specified points.

  3. Horizontal Asymptotes

    • Find horizontal asymptotes of various functions.

Lab Assignment 3

  1. Polar Coordinates

    • Determine polar coordinates and convert between forms.

  2. Polar Curve Analysis

    • Identify values leading to intersections and curve characteristics.

  3. Function Behavior

    • Calculate limits and analyze the behavior of given functions at specified points.

Lab Assignment 2

  1. Implicit and Explicit Forms

    • Understand and convert between implicit and explicit functions.

  2. Parametric Curve Analysis

    • Find Cartesian equations from parametric forms.

  3. Level Curves in Different Forms

    • Express level curves in various forms (explicit, parametric).

Lab Assignment 1

  1. Logarithmic Equations

    • Solve and verify a logarithmic equation.

  2. Hyperbolic Functions and Identities

    • Identify identities and solve hyperbolic equations.

  3. Market Analysis of Functions

    • Discuss the importance of expressing values exactly rather than approximately.

General Notes

  • Review limits, discontinuities, and curves.

  • Consider the relationship between different forms of functions and their derivatives.

  • Prepare for practical applications in optimization and physical modeling.