AP Calculus BC Comprehensive Syllabus

When this interval is taken to be infinitesimally small, we get the instantaneous rate of change at a particular point, which is the derivative of the function at that point. It is the slope of the tangent line to the graph of the function at that point.

Limits: A limit is a value that a function approaches as the input variable approaches a certain value. 

There are various methods to represent limits, including algebraic manipulation, factoring, rationalizing, and applying trigonometric or logarithmic identities.

• Domain: The set of all possible input values for a function.
• Range: The set of all possible output values for a function.
• Intercepts: The x-intercept is where the graph of a function crosses the x-axis, while the y-intercept is where the graph crosses the y-axis.
• Symmetry: A function is symmetric if its graph can be reflected over a certain line and still look the same.
• Continuity: A function is continuous if there are no abrupt breaks or holes in its graph. This means that the function can be drawn without lifting your pencil from the paper.
• Points of discontinuity: Points where a function is not continuous, either because the function has a hole or a jump.
• Local extrema: Points on a function where the function has a maximum or minimum value in a small interval around that point.
• Concavity: A function is concave up if its graph curves upward, and concave down if its graph curves downward. The point where the concavity changes is called an inflection point.

Limits do not exist at a point when the function approaches different values from the left and right sides of that point, or when the function approaches infinity or negative infinity at that point.

1. X values: The values of x in the table, which approach the limit value.
2. f(x) values: The corresponding function values for each x-value in the table.

Limit theorems are rules that help us evaluate limits of functions based on the behavior of the function at nearby points. 

Using limit theorems, we can find the limits of sums, differences, products, quotients, and composite functions. These theorems include: 

• Squeeze Theorem
• Algebraic Limit Theorem
• Limit of a Composite Function Theorem

To use equivalent expressions to evaluate a limit, you can manipulate the original function algebraically to create an equivalent expression that is easier to evaluate. 

Some common techniques for simplifying functions include

• Factoring
• Rationalizing the denominator
• Using algebraic identities.

1. Direct substitution: This method involves substituting the value at which the function is approaching directly into the function to determine the limit. 
2. Factorization: This method involves factoring the numerator and denominator of the function and canceling out common factors to simplify the function. 
3. Conjugate method: This method involves multiplying the numerator and denominator of the function by the conjugate of the denominator to eliminate square roots or radicals in the denominator. 
4. L’Hopital’s Rule: This method involves taking the derivative of both the numerator and denominator of the function and evaluating the limit of the resulting ratio. If the resulting limit is of the form “0/0” or “infinity/infinity”, then this method can be used to evaluate the limit.
5. Squeeze theorem: This method involves sandwiching the function between two other functions that have the same limit. If the limits of the two other functions are equal, then the limit of the original function is also equal to this limit.

Types of discontinuities:

1. Removable discontinuity: A removable discontinuity occurs when a function has a hole at a certain point. The function is undefined at that point, but it can be made continuous by defining the function at that point.
2. Jump discontinuity: A jump discontinuity occurs when the function has a sudden jump in value at a certain point. The left-hand and right-hand limits exist, but they are not equal.
3. Infinite discontinuity: An infinite discontinuity occurs when the function approaches infinity or negative infinity at a certain point. The limit of the function does not exist at that point.
4. Oscillating discontinuity: An oscillating discontinuity occurs when the function oscillates between two values infinitely many times as it approaches a certain point. The limit of the function does not exist at that point.

1. The function is defined at x = a.
2. The limit of the function as x approaches a from both the left and the right exists.
3. The value of the function at x = a is equal to the limit.

1. Taking the limit of the function as it approaches the point of discontinuity
2. Simplifying the function by factoring or canceling out common factors
3. Re-defining the function at the point of discontinuity with a new value that makes the function continuous
4. Applying a piecewise function to the original function, with different rules for the left and right limits at the point of discontinuity

To determine the values of x that make a discontinuous function continuous:

1. Identify the point(s) of discontinuity of the function
2. Determine the type of discontinuity at each point (e.g. removable, jump, infinite)
3. Apply one of the techniques mentioned above to remove the discontinuity at each point
4. Check if the function is now continuous at the previously discontinuous point(s)
5. If the function is still discontinuous, repeat the process until it becomes continuous at all points

Infinite limits: An infinite limit occurs when the limit of a function as the input variable approaches a certain value (usually infinity or negative infinity) does not exist because the function grows or decays without bound. 

Asymptotic behavior: Asymptotic behavior refers to the behavior of a function as the input variable approaches infinity or negative infinity. A function can have a horizontal asymptote, which is a straight line that the function approaches but never touches. 

Unbounded behavior: Unbounded behavior refers to the behavior of a function that grows or decays without bound as the input variable approaches a certain value (usually infinity or negative infinity). 

When comparing the relative magnitudes of functions and their rates of change, we can use limits to determine which function grows faster or slower than another function. 

The behavior of a function on an interval refers to how the function behaves or changes within that interval. 

The difference quotients f(a+h) – f(a)h and f(x) – f(a)x-a express the average rate of change of a function over an interval.

The instantaneous rate of change refers to the rate of change of a function at a specific point. It is the limit of the average rate of change as the interval around the point shrinks to zero.

It is expressed by: h0f(a+h) – f(a)h or xaf(x) – f(a)x-a provided the limit exists.

For y = f(x), the derivative is dydx, f'(x) and y’

The derivative of a function at a point is the slope of the tangent line to the graph of the function at that point. 

There are several reasons why a function may not be differentiable at a point, including:

• The function has a sharp corner or cusp at that point.
• The function has a vertical tangent (i.e., the slope of the tangent line approaches infinity or negative infinity) at that point.
• The function has a discontinuity (either a jump or an infinite oscillation) at that point.

1. Constant Rule: The derivative of a constant is zero.
2. Sum Rule: The derivative of the sum of two functions is equal to the sum of their individual derivatives.
3. Difference Rule: The derivative of the difference between two functions is equal to the difference of their individual derivatives.
4. Constant Multiple Rule: The derivative of a constant times a function is equal to the constant times the derivative of the function.

These rules are essential in finding the derivative of more complex functions and can be combined with other techniques such as the product rule, quotient rule, and chain rule to calculate derivatives of more complicated functions.

Derivative of cos(x): ddx(cos(x)) = -sin(x)

Derivative of sin(x): ddx(sin(x)) = cos(x)

Derivative of ex: ddx(ex) = ex

Derivative of ln(x): ddx(ln(x)) = 1/x

ddx(u(x) v(x)) = u(x) (ddx)v(x) + v(x) (ddx)u(x)

To take the derivative of a product of two functions, you multiply the first function by the derivative of the second function, and add to it the product of the second function and the derivative of the first function.

(ddx)(u(x)v(x)) = [v(x) (ddx)u(x) – u(x) (ddx)v(x)] / [v(x)]2

To take the derivative of a quotient of two functions, you subtract the product of the first function and the derivative of the second function from the product of the second function and the derivative of the first function, and divide the result by the square of the second function.

• The derivative of the tangent function, tan(x), is sec2(x)
• The derivative of the cotangent function, cot(x), is -csc2(x)
• The derivative of the secant function, sec(x), is sec(x) tan(x)
• The derivative of the cosecant function, csc(x), is -csc(x) cot(x)

To apply these rules, you’ll need to be familiar with the basic trigonometric identities, such as

•  sin2(x) +  cos2(x) = 1

and the reciprocal identities, such as 

• sec(x) = 1/cos(x).

If a function f(x) can be written as a composite function g(h(x)), where g is a function of one variable and h is a function of x, then the chain rule states that the derivative of f(x) is given by:

(f(g(h(x))))’ = f'(g(h(x))) g'(h(x)) h'(x)

In other words, to take the derivative of a composite function, you first find the derivative of the outermost function, evaluated at the inner function. Then you multiply that by the derivative of the inner function, and finally multiply that by the derivative of the input variable.

An implicitly defined function is a function that is not given in the form y = f(x), but rather in a more complicated equation involving both x and y.

To find the derivative of an implicitly defined function, we differentiate both sides of the equation with respect to x, treating y as a function of x. This means that we use the chain rule to differentiate any terms that involve y.

To differentiate an inverse function f-1(x), we use the formula: (f-1)'(x) = 1 / f'(f-1(x))

• arcsin (or sin-1) = 1(1 – x2
• arccos (or cos-1) = – 1(1 – x2
• arctan (or tan-1) = 1(1 + x2)

The second derivative of a function f(x) is the derivative of its first derivative. It can be represented using the following notations: f”(x) = d²y/dx² = (d/dx) (dy/dx)

The second derivative measures the rate of change of the slope of the function with respect to the independent variable x. 

This concept can be applied to various real-world situations to express information such as velocity, acceleration, and optimization. 

For example, the derivative of a position function can give us information about an object’s velocity, and the second derivative can tell us about its acceleration.

To understand how derivatives can be used to solve rectilinear motion problems, we can connect position, velocity, and acceleration through differentiation. 

• The position of an object at any time t can be represented by a function s(t). 
• The derivative of s(t) with respect to time t gives us the velocity of the object, which we can represent as v(t) = s'(t). 
• The second derivative of s(t) with respect to time t gives us the acceleration of the object, which we can represent as a(t) = s”(t).

1. Population Growth: For instance, the rate of change of a population can be used to determine how quickly a disease will spread within a community.
2. Finance: Rate of change of an investment’s value over time, or the rate of change of a company’s profits.
3. Chemistry: To model chemical reactions and determine the rate at which reactants are consumed and products are formed.
4. Biology: To model biological processes, such as the rate at which a tumor grows or the rate at which an enzyme catalyzes a reaction.
5. Engineering: For modeling and optimizing processes, such as the rate at which heat is transferred in a system.
6. Economics: To model economic scenarios, such as the rate of change of a product’s demand.

To solve related rates problems, there are several rules and techniques that can be used:

1. Implicit differentiation: This technique is used when the relationship between the variables is not given explicitly, but rather implicitly in an equation. 
2. Chain rule: The chain rule is used when we have a composite function.
3. Product rule: The product rule is used when we have a product of two variables that are both changing with respect to time.
4. Quotient rule: The quotient rule is used when we have a quotient of two variables that are both changing with respect to time. 
5. Trigonometric identities: In some problems, trigonometric identities can be used to simplify expressions and make them easier to differentiate.

Linearisation is the process of finding this linear approximation of a function around a given point. The linearisation of a function f(x) at a point x=a is the linear function L(x) = f(a) + f'(a)(x-a), where f'(a) is the derivative of f(x) evaluated at x=a.

Using the equation of the tangent line, we can approximate the value of a function at a point nearby to where the tangent line intersects the curve. To do this, we evaluate the tangent line equation at the desired point, which gives an approximation of the function value at that point.

L’Hospital’s Rule is a technique used to evaluate limits of indeterminate forms. The rule states that if we have a limit of the form 0/0 or infinity/infinity, we can take the derivative of both the numerator and denominator with respect to the variable, and then evaluate the limit again. This process can be repeated as many times as necessary until we reach a limit that is no longer indeterminate.

The formal statement of L’Hospital’s Rule is:

If f(x) and g(x) are differentiable functions such that xaf(x)=0 and xag(x)=0 or xaf(x)=±∞  and xag(x)=±∞  then:

xaf(x)g(x) =xaf'(x)g'(x)  

provided that the limit on the right-hand side exists or is infinite.

Application of MVT: One common application is to use the MVT to find the average rate of change of a function over an interval when only the values of the function at the endpoints of the interval are known. 

Critical point of a function: A critical point of a function is a point where the derivative of the function is either zero or undefined. At a critical point, the slope of the tangent line to the graph of the function is either zero or does not exist.

Global versus Local Extrema: Global extrema refers to the highest or lowest values of a function over its entire domain or over a specific interval. Local extrema, on the other hand, refers to the highest or lowest values of a function in a particular region or interval, & can be found at critical points. 

The first derivative of a function provides information about the behavior of the function in terms of its increasing and decreasing intervals. Positive first derivatives mean the function is increasing at that point.

The first derivative helps to identify the intervals of the function where it is increasing, decreasing, or having a critical point. 

The first derivative test is a method used to find relative local extrema of a function. It involves taking the derivative of the function and analyzing its sign in intervals around critical points, which are points where the derivative is equal to zero or undefined.

A point of inflection on a graph is where the concavity of the graph changes from upward to downward, or vice versa. 

The second derivative of a function can be used to identify these points of inflection and determine the intervals of upward or downward concavity for the function’s graph.

The Second Derivative Test states that if the second derivative of a function evaluated at a critical point is:

• Positive, the function has a local minimum at that point. 
• Negative, the function has a local maximum at that point. 
• Zero, the Second Derivative Test is inconclusive, and we need to use additional methods to determine the nature of the critical point.

Graphs of Functions:

• Domain and Range: the set of all possible inputs and outputs of the function, respectively
• Intercepts: points where the graph intersects with the x- or y-axis
• Symmetry: whether the graph is symmetric with respect to the x-axis, y-axis, or origin
• Asymptotes: lines that the graph approaches but never touches
• Extrema: maximum and minimum points on the graph
• Inflection points: points where the concavity of the graph changes

Graphs of Derivatives:

• Domain: the set of all possible inputs for the derivative function
• Increasing and Decreasing Intervals: intervals where the function is increasing or decreasing, respectively
• Local Extrema: points where the derivative is equal to zero and changes sign (i.e., from positive to negative or vice versa)
• Concavity: whether the function is concave up or down
• Inflection points: points where the concavity of the graph changes

Together, these derivatives can help us understand the behavior of a function in great detail.

For minimum values, example: to find the minimum cost of producing a certain number of items, we would need to use calculus to find the minimum point of the cost function, where the derivative is equal to zero. This minimum point would represent the lowest possible cost that can be achieved for producing that number of items.

Some common behaviors of implicit relations that are studied in AP Calculus BC include:

1. Finding the slope: The slope of an implicit relation can be determined by taking the derivative of both sides of the equation with respect to one of the variables.
2. Finding the concavity: The concavity of an implicit relation can be determined by taking the second derivative of both sides of the equation with respect to one of the variables.
3. Finding critical points: Critical points of an implicit relation can be found by solving the equation for when the first derivative is equal to zero.
4. Finding inflection points: Inflection points of an implicit relation can be found by solving the equation for when the second derivative changes sign.
5. Sketching the curve: By using the above techniques and the behavior of the curve at the critical points and inflection points, a sketch of the curve can be made.

The second derivative can be used to analyze the concavity and inflection points of the original function. Second derivatives involving implicit differentiation refer to the process of finding the second derivative of a function that is defined implicitly by an equation involving both x and y variables.

The positive accumulation of change refers to the total amount by which a function increases over a given interval. It is calculated by finding the definite integral of the function over the interval, where the function values are positive.

The negative accumulation of change refers to the total amount by which a function decreases over a given interval. It is calculated by finding the definite integral of the function over the interval, where the function values are negative.

Approximating definite integrals: The most common methods used are the trapezoidal rule, Simpson’s rule, and the midpoint rule. These methods involve dividing the interval of integration into smaller subintervals, and then using the values of the function at the endpoints of these subintervals to estimate the area.

Riemann’s sums are a way to approximate the value of a definite integral by dividing the interval of integration into subintervals and then approximating the area of each subinterval using a rectangle. The sum of these rectangle areas gives an estimate of the area under the curve of the function being integrated. 

Definite integral notation is used to represent the area under a curve between two points on the x-axis. It is denoted by the symbol ∫, and the expression to be integrated, with the integration limits (upper and lower bounds of integration) written as subscripts and superscripts.

Let f(x) be a continuous function on an interval [a,b]. The accumulation function F(x) is defined as:

F(x) = [a,x] f(t) dt 

In AP Calculus BC, definite integrals have several properties that are essential to understand.

1. Constant multiple rule: The integral of a constant times a function is equal to the constant times the integral of the function. In other words, for any constant c, c f(x) dx =cf(x) dx
2. Sum and difference rules: The integral of the sum (or difference) of two functions is equal to the sum (or difference) of the integrals of the two functions. In other words, [f(x) + g(x)] dx =f(x)dx + g(x)dx , and [f(x) – g(x)] dx =f(x)dx – g(x)dx
3. Reversal of limits of integration: When the limits of integration are reversed, the sign of the integral changes. In other words, if a and b are the limits of integration, then abf(x)dx = – baf(x)dx
4. Integration over adjacent intervals: The integral of a function over an interval can be split into the sum of integrals over adjacent subintervals. In other words, if c is a point in the interval [a,b], thenabf(x)dx =  acf(x)dx +cbf(x)dx

To evaluate a definite integral using antiderivatives and the Fundamental Theorem of Calculus, you first find an antiderivative of the integrand, and then evaluate that antiderivative at the limits of integration and subtract the results. This process is often called “evaluating the integral by antiderivatives.”

Indefinite integral, also called an antiderivative, is the family of functions that differ by a constant that can be obtained by integrating a given function. It is written using the integral symbol and the function being integrated.

Differentiation rules are used to find the derivative of a function. Some common rules include the power rule, product rule, quotient rule, chain rule, and trigonometric rules.

Closed-form antiderivatives are functions that can be expressed using a finite number of elementary functions, such as polynomials, exponential functions, logarithmic functions, and trigonometric functions. However, not all functions have closed-form antiderivatives, and in these cases, numerical methods or approximation techniques may be used to approximate the value of the integral.

Integration by substitution can also be used for definite integrals. In this case, the limits of integration must be adjusted to account for the change in variable. The new limits are found by substituting the original limits into the substitution equation.

The formula for integration by substitution is:

f(g(x)) g'(x) dx =  f(u)du  (where u = g(x)) 

This formula states that if we can write the integrand as f(g(x)) g'(x), we can make the substitution u = g(x) and rewrite the integral in terms of u.

A rational function is written in the form:

f(x) = P(x) / Q(x)

where P(x) and Q(x) are polynomials and Q(x) is not the zero polynomial.

Specifically, an improper integral is a limit of a definite integral as one or both of the limits of integration become infinite or as one or more points within the interval of integration approach a singular point (such as a vertical asymptote).

An improper integral can be expressed as:

a∞f(x) dx =t∞atf(x) dx   or

a∞ f(x) dx =cb–acf(x) dx+da+dbf(x) dx

where “a” and “b” are finite or infinite limits of integration and “f(x)” is the function being integrated. The limits “t”, “c”, and “d” approach infinity or singular points within the interval of integration.

Improper integrals can be evaluated using various techniques, such as comparison test, limit comparison test, integral test, or the convergence/divergence test.

1. Integration by substitution
2. Integration by parts
3. Partial fraction decomposition
4. Trigonometric substitution
5. Improper integration

They relate a function of an independent variable and the function’s derivatives.

To find the general solution of a differential equation, we use anti-differentiation to obtain an expression that contains an arbitrary constant (known as the constant of integration). This expression represents the family of all possible solutions to the differential equation.

Specific applications of finding general and particular solutions to differential equations, such as exponential growth and decay models which are commonly used in various fields including biology, economics, and physics will be studied.

The purpose of the limiting value, or carrying capacity, in this logistic differential equation is to represent the maximum sustainable population size that can be supported by the available resources in a given environment. 

The average value of a continuous function f on an interval [a, b] in AP Calculus BC is given by the formula:

avg (f) = 1(b-a)abf(x) dx

This formula calculates the mean value of f over the interval [a, b] and represents the height of a horizontal line that would have the same area as the region bounded by the graph of f, the x-axis, and the vertical lines x = a and x = b.

The definite integral of velocity in rectilinear motion represents the displacement of an object over a given time interval. If v(t) represents the velocity of an object at time t, then the displacement of the object over the interval [a, b] is given by the definite integral:

abv(t) dt

The definite integral of the rate of change of a quantity represents the total change in that quantity over a given interval. For example, the definite integral of the rate of change of a population over a time interval represents the total change in the population during that interval.

For a function f(x) with limits a and b, the volume of the solid generated by revolving the region bounded by the graph of f(x), the x-axis, and the vertical lines x = a and x = b around the x-axis is given by:

V=π ab(f(x))2 dx

Similarly, for a function g(y) with limits c and d, the volume of the solid generated by revolving the region bounded by the graph of g(y), the y-axis, and the horizontal lines y = c and y = d around the y-axis is given by:

V=π cd(g(y))2 dy

In this case, the radius of the disc is a function of the distance from the axis of revolution. For example, if the solid is generated by revolving the region bounded by y = f(x), y = g(x), x = a, and x = b around the line y = k, then the volume is given by:

V=π ab(f(x)-k)2–(g(x)-k)2 dx

The washer method is used to find the volume of a solid generated by revolving a region bounded by two curves around the x or y-axis. For example, if the region is bounded by y = f(x), y = g(x), x = a, and x = b, and is revolved around the x-axis, then the volume is given by:

V=π ab[(f(x))2–(g(x))2 dx]

Similarly, if the region is bounded by x = f(y), x = g(y), y = c, and y = d, and is revolved around the y-axis, then the volume is given by:

V=π cd[(f(y))2–(g(y))2] dx

If the solid is generated by revolving the region bounded by y = f(x), y = g(x), x = a, and x = b around the line y = k, then the volume is given by:

V=π ab[(f(x)-k)2–(g(x)-k)2] dx

The length of the curve is then found by integrating the derivative over the domain of the curve.

The arc length of a smooth, planar curve is given by the formula:

L=ab1+dydx2 dx

where a and b are the limits of integration. The distance traveled by an object moving along the curve is also given by this formula. If the curve is given by the equation y = f(x), then the formula can be rewritten as:

L=ab1+(f'(x))2 dx

where f'(x) is the derivative of f(x) with respect to x.

The general form of a parametric function is:

x = f(t)

y = g(t)

where x and y are the coordinates of a point on the curve and t is the parameter.

To find the derivative of a parametric function, we use the chain rule. The derivative of the x-coordinate function is:

dx/dt = f'(t)

And the derivative of the y-coordinate function is:

dy/dt = g'(t)

d2xdt2 = f”(t)   and    d2ydt2 = g”(t)

L=abdxdt2+dydt2 dt

where a and b are the limits of the parameter t that correspond to the start and end points of the curve. This formula is derived from the Pythagorean theorem.

To differentiate a vector-valued function, you differentiate each component of the vector separately using the rules of differentiation. This will give you the derivative of the vector-valued function, which represents the tangent vector to the curve at any point on the curve.

Differentiation in polar form involves finding the derivative of a polar function, which requires applying the chain rule and the product rule in a modified form.

Polar equations are equations that relate polar coordinates to each other, such as:

r = 2cos(θ) or r2 = 4 sin(2θ).

The graph of a polar equation is called a polar curve.

A =12 (f(θ))2 dθ

where f(θ) is the polar function defining the boundary of the region. 

A = 12 (g(θ))2 – (f(θ))2 dθ

where f(θ) and g(θ) are the two polar functions defining the boundaries of the region.

Divergent infinite series: A series is said to diverge if the sequence of its partial sums does not approach a finite limit as the number of terms in the series approaches infinity.

Determining convergence or divergence: There are various tests that can be used to determine whether a series converges or diverges, including the ratio test, the root test, the integral test, and the comparison test.

nth partial sum: The nth partial sum of a series is the sum of the first n terms of the series.

Condition for geometric series: A geometric series converges if and only if the absolute value of the common ratio is less than 1. If the absolute value of the common ratio is greater than or equal to 1, the series diverges.

The p-series is a series of the form 1/n^p, where p is a positive constant. If p > 1, then the series converges, but if p ≤ 1, then the series diverges.

This theorem is useful for estimating the error in approximating series sums.

Pn(x) = f(a) + f'(a)(x-a) + (f”(a)/2!)(x-a)^2 + … + (f^(n)(a)/n!)(x-a)^n

where f^(n) denotes the nth derivative of f. The Taylor polynomial approximates the function f to a certain degree of accuracy around the point a.

Rn(x) = (f(n+1)(c)/(n+1)!)(x-a)(n+1)

where c is some value between x and a. 

The Lagrange Error Bound states that the absolute value of the error between f(x) and its nth-degree Taylor polynomial is less than or equal to the absolute value of the maximum value of the (n+1)th derivative of f on the interval between x and a, multiplied by the distance between x and a, raised to the (n+1)st power, divided by (n+1)! 

This theorem is useful for estimating the error in approximating a function using its Taylor polynomial.

The interval of convergence of a power series is the set of all x-values for which the series converges. The interval of convergence can be determined using a variety of convergence tests, including the ratio test, the root test, and the alternating series test.

A power series is an infinite series of the form ∑(n=0 to infinity) cn(x-a)n, where a is the center of the series and the coefficients cn are constants. The variable x is the independent variable, and the power series can be used to represent a function as an infinite sum of powers of x.

A converging power series is a power series that converges to a finite value for at least some values of x. A power series may converge for some values of x and diverge for others.

The ratio test is a convergence test that can be used to determine the convergence or divergence of a power series. The ratio test involves taking the limit of the absolute value of the ratio of successive terms of the series as n approaches infinity.

A Maclaurin series is a special case of a Taylor series where the center of the series is at x=0. The coefficients of the series are determined using the function’s derivatives evaluated at x=0. Maclaurin series can be used to approximate a function near x=0.

Both Taylor and Maclaurin series can be used to approximate a function near a particular point. The accuracy of the approximation depends on the number of terms included in the series. 

The Maclaurin series for sine, cosine, and tangent are as follows:

sin(x) = x – x^3/3! + x^5/5! – x^7/7! + …

cos(x) = 1 – x^2/2! + x^4/4! – x^6/6! + …

tan(x) = x + x^3/3 + 2x^5/15 + 17x^7/315 + …

These series can be used to approximate sine, cosine, and tangent functions near x=0.