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3.10

Linear Approximations and Differentials

Linear Approximations and Differentials

$\frac{dy}{dt}=ky$

This equation is called the Law of Natural Growth (if k > 0) or The Law of Natural Decay (if k < 0). It is called a differential equation because it involves an unknown function y and and its derivative dy/dt

Exponential Growth and Decay

$y\left(t\right) = y\left(0\right)e^{kt}$

The only sollution of the differential equation dy/dt = ky are the exponential functions above.

Population Growth

Radioactive Decay

Newton's Law of Cooling

Continuously Compounded Interest

Links

• Home
• 3.1: Derivatives of Polynomials and Exponential Functions
• 3.2: The Product and Quotient Rules
• 3.3: Derivatives of Trigonometric Functions
• 3.4: The Chain Rule
• 3.5: Implicit Differentiation
• 3.6: Steps in Logarithmic Differentiation
• 3.7: Rates of Change in the Natural and Social Sciences
• 3.8: Exponential Growth and Decay
• 3.9: Related Rates
• 3.10: Linear Approximations and Differentials
• 3.11: Hyperbolic Functions