TExES Mathematics Differential Calculus and Derivatives Quiz

To find the derivative of the function f(x) = 3x² + 2x - 5, apply the power rule. The derivative of 3x² is 6x, the derivative of 2x is 2, and the derivative of a constant (-5) is 0. Combining these results gives the derivative as 6x + 2.

The limit definition lim(h→0) [f(x+h) - f(x)]/h calculates the instantaneous rate of change of the function f(x) at a specific point x. This concept is fundamental in calculus, representing the derivative, which describes how the function behaves locally around that point.

To find the derivative of g(x) = e^x · cos(x), apply the product rule, which states that (uv)' = u'v + uv'. Here, u = e^x (derivative is e^x) and v = cos(x) (derivative is -sin(x)). Thus, g'(x) = e^x · cos(x) + e^x · (-sin(x)), simplifying to e^x · (cos(x) - sin(x)).

To find the derivative of h(x) = (2x³ + 1)⁵, the Chain Rule is most efficient because it allows us to differentiate composite functions. Here, the outer function is raised to the fifth power, and the inner function is 2x³ + 1. Applying the Chain Rule simplifies the differentiation process effectively.

The derivative of the natural logarithm function, f(x) = ln(x), is derived using the limit definition of the derivative. It measures the rate of change of the function. The result, 1/x, indicates that as x increases, the slope of the tangent line to the curve decreases, reflecting the logarithmic growth behavior.

A critical point occurs where the derivative of a function is zero or undefined, indicating potential changes in the function's behavior. Specifically, it helps identify local maxima and minima, where the function reaches its highest or lowest values in a given interval, thus playing a crucial role in understanding the function's overall shape.

To classify the critical point at \( x = \sqrt{4/3} \), we first find the second derivative of \( f(x) \). Evaluating the second derivative at this point yields a positive value, indicating that the function is concave up. Thus, the critical point corresponds to a local minimum.

The second derivative, f''(x), provides information about the curvature of a function's graph. When f''(x) is positive, the graph is concave up, indicating that the slope of the tangent line is increasing. Conversely, when f''(x) is negative, the graph is concave down, showing that the slope is decreasing.

To find f'(x) using the quotient rule, we differentiate the numerator and denominator separately. The numerator's derivative is 2x, and the denominator's derivative is 1. Applying the quotient rule formula, we combine these results, leading to the expression (2x(x + 2) - (x² - 1))/(x + 2)², which represents the rate of change of f(x).

A constant function does not change regardless of the input value. Since the derivative measures the rate of change, and a constant function has no change, its derivative is always zero. This property holds for any constant value, confirming that the derivative of a constant function is indeed zero.

The derivative of the sine function, f(x) = sin(x), is the cosine function. This is a fundamental result from calculus, which states that the rate of change of the sine function at any point x is given by the cosine of that point. Thus, f'(x) = cos(x).

A function is considered increasing on an interval when its derivative, f'(x), is greater than zero. This indicates that as the input values (x) increase, the output values (f(x)) also increase, reflecting a positive slope on the graph of the function within that interval.