Calculus Concepts: Limits and Derivatives

The rate of change of a function's output with respect to its input.

The process of finding the derivative of a function.

A function's rate of change at a specific point.

The rate at which an object's velocity changes.

The interval where an object's velocity is increasing.

The interval where an object's velocity is decreasing.

The rate at which a variable changes over time.

The volume of a sphere. Formula: V = (4/3)πr^3

The derivative of a function represents the instantaneous rate of change of the function's output with respect to its input. It describes how much the output changes for a tiny change in the input.

A horizontal tangent line occurs on a function's graph where the derivative is zero. This means the function's output is not changing momentarily.

To find the x-coordinates where a function has a horizontal tangent, set the derivative of the function equal to zero and solve for x.

When differentiating a function, every term with an x must be treated individually. The power rule is applied for each term, and the constant term disappears.

The power rule states that the derivative of x raised to a power 'n' is 'n' times x raised to the power 'n-1'.

To find the x-coordinates where a function has a horizontal tangent, we need to differentiate the function and solve the resulting equation for x.

The equation must first be differentiated with respect to x. Then, the derivative is set to zero and the resulting equation is solved for x.

Horizontal tangents occur where the derivative of the function is equal to zero. Finding these points gives us key insights into a function's behavior.

The integral of f(x) with respect to x, where the limits of integration are from f(x) to f(x). It represents the area under the curve of f(x) between the points f(x) and x.

Represents the integral of the composite function f(g(x)) with respect to x, where the limits of integration are from g(x) to g(x). This signifies the area beneath the composite function's curve from g(x) to x.

Represents the integral of the function f(x) with respect to x, where the limits of integration are from f(x) to f(x) multiplied by 2. It represents the area under the curve of f(x) between the points f(x) and 2*f(x).

Represents the integral of f(x) with respect to x, where the limits of integration are from f(x) to f(x). It signifies the area under the curve of f(x) between the points f(x) and x.

Euler's method is a numerical method for approximating solutions to differential equations. It works by using the derivative at a given point to estimate the value of the function at a nearby point.

A step size in Euler's method is the increment of the independent variable used in each step of the calculation. It determines the precision of the approximation.

The carrying capacity in a logistic differential equation represents the maximum population size that an environment can sustain under ideal conditions.

A logistic differential equation is a mathematical model used to describe population growth that is limited by resources. It takes into account the carrying capacity of the environment.

A removable discontinuity occurs at a point where the function has a hole, but the limit exists as x approaches that point.

A non-removable discontinuity occurs when a function has a break or jump at a specific point, and the limit does not exist as x approaches that point.

Finding the total area of a region bounded by curves involves breaking the region into smaller parts and calculating the area of each part. When integrating, the function on top determines the upper limit and the function on the bottom determines the lower limit.

To find the total area enclosed by two curves, integrate the difference between the top function and the bottom function over the interval of the region.

The region R is bound by the x-axis, the y-axis, the curve y = e^x, and the vertical line x = 1. This region lies entirely in the first quadrant and is shaped like a wedge.

Points of inflection occur where the concavity of a function changes. On the graph, this looks like a transition from a curve facing upwards to a curve facing downwards, or vice versa.

The x-coordinates of the points of inflection can be found by setting the second derivative of the function equal to zero and solving for x.

The interval where a function's second derivative is positive indicates that the function is concave up.

The interval where a function's second derivative is negative indicates that the function is concave down.

The integral of a function's acceleration with respect to time gives us the velocity function.

The integral of a function's velocity with respect to time gives us the position function.

The derivative of a distance function gives us the velocity function. The derivative of velocity gives us the acceleration function.

The position of a particle at any time can be found by integrating its velocity function. The initial position (position at time t=0) is often provided to help determine the constant of integration.

Finding the derivative of an inverse function is done by applying the inverse function theorem, which states that the derivative of the inverse function at a point is the reciprocal of the derivative of the original function at the corresponding point. To find (F-1)'(7), we first need to find the value of x for which F(x) = 7.

The total distance traveled by a particle is given by the integral of the absolute value of its velocity function over the given time interval. In this case, we need to integrate the absolute value of the velocity function, v(t) = -3t, from t = 0 to t = 4.

To approximate the number of reindeer after 5 years, we need to integrate the growth rate function, P(t), from t = 0 to t = 5 and add this to the initial population of 100 reindeer.

Evaluate the definite integral using appropriate substitution techniques to simplify the expression. Pay attention to the limits of integration and the resulting antiderivative.

Determine the intervals on which the function is increasing, decreasing, and the coordinates of relative minima and maxima. These features are found through the analysis of the first and second derivatives.

A function is increasing on an interval where the first derivative is positive, decreasing on an interval where the first derivative is negative.

Relative extrema occur at critical points where the first derivative changes sign. A relative maximum occurs where the derivative changes from positive to negative, while a relative minimum occurs where the derivative changes from negative to positive.

Concavity is determined by analyzing the sign of the second derivative. A function is concave up on an interval where the second derivative is positive and concave down where the second derivative is negative.