Final answer:
The derivatives of functions f(x) = x^2, g(x) = sin(x), h(x) = e^x, and k(x) = ln(x) can be found using the definition of derivative. The slope of the tangent line and the instantaneous rates of change correspond to the value of these derivatives at the given points.
Step-by-step explanation:
Derivatives Using the Definition
Derivative is a fundamental concept in calculus, providing a way to find the rate of change of a function. Derivatives are often calculated using limits according to the definition of the derivative.
- (a) For the function f(x) = x^2, the derivative is found by taking the limit as h approaches 0 of the difference quotient: (f(x+h) - f(x)) / h. This gives us 2x after simplification.
- (b) The derivative of g(x) = sin(x) is found similarly using the difference quotient and simplifying to reveal the derivative, which is cos(x).
- (c) To find the slope of the tangent line to h(x) = e^x at x=3, we use the derivative of e^x which is e^x itself, thus at x=3, the slope is e^3.
- (d) The instantaneous rate of change for k(x) = ln(x) at x = 1 is given by the derivative 1/x, which is 1 at x = 1.
Using a calculator can greatly assist in exploring these concepts further and verifying the results obtained through the definition.