Calculus reference sheet for students

Calculus Cheat Sheet

Integrals

The integral of a function \(f(x)\) from \(a\) to \(b\) is given by: \[\int_{{a}}^{{b}} f(x) \, dx\]

Power Rule: \[\int x^n \, dx = \frac{{x^{n+1}}}{{n+1}} + C\] where \(C\) is the constant of integration.

Constant Multiple Rule: \[\int cf(x) \, dx = c \cdot \int f(x) \, dx\]

Sum Rule: \[\int [f(x) + g(x)] \, dx = \int f(x) \, dx + \int g(x) \, dx\]

Difference Rule: \[\int [f(x) – g(x)] \, dx = \int f(x) \, dx – \int g(x) \, dx\]

Exponential Function: \[\int e^x \, dx = e^x + C\]

Natural Logarithm: \[\int \ln(x) \, dx = x \ln(x) – x + C\]

Trigonometric Functions: \[\int \sin(x) \, dx = -\cos(x) + C\] \[\int \cos(x) \, dx = \sin(x) + C\] \[\int \tan(x) \, dx = -\ln|\cos(x)| + C\]

Inverse Trigonometric Functions: \[\int \frac{1}{\sqrt{1-x^2}} \, dx = \arcsin(x) + C\] \[\int -\frac{1}{\sqrt{1-x^2}} \, dx = \arccos(x) + C\] \[\int \frac{1}{1+x^2} \, dx = \arctan(x) + C\]

Integration by Parts: \[\int u \, dv = uv – \int v \, du\]

Trigonometric Substitution: \[\int \frac{1}{\sqrt{a^2 – x^2}} \, dx = \arcsin\left(\frac{x}{a}\right) + C\] \[\int \frac{1}{a^2 + x^2} \, dx = \frac{1}{a} \arctan\left(\frac{x}{a}\right) + C\]

Fundamental Theorem of Calculus and Getting More from an Integral Table

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus establishes a fundamental link between differentiation and integration. It states that if \(F(x)\) is an antiderivative (indefinite integral) of \(f(x)\) (i.e., \(\frac{{dF(x)}}{{dx}} = f(x)\)), then the definite integral of \(f(x)\) from \(a\) to \(b\) is equal to \(F(b) – F(a)\): \[\int_{{a}}^{{b}} f(x) \, dx = F(b) – F(a)\]

Using the Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus allows us to find definite integrals by evaluating antiderivatives at the upper and lower limits of integration. It enables us to compute the area under a curve and evaluate definite integrals without resorting to Riemann sums or other numerical methods.

However, the theorem can also be used in reverse to help us find antiderivatives (indefinite integrals) of functions. If we know the derivative of a function, we can find its antiderivative by integrating the derivative with respect to \(x\). This is how we obtain integral tables, which list common antiderivatives and their corresponding derivatives.

Using the Integral Table for Differentiation of Integrals

One useful application of an integral table is finding the derivatives of integrals. Given an integral with varying limits, such as: \[F(x) = \int_{{a(x)}}^{{b(x)}} f(t) \, dt\] where \(a(x)\) and \(b(x)\) are functions of \(x\), we can differentiate it with respect to \(x\) using the Fundamental Theorem of Calculus.

The result is obtained by applying the chain rule, which states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. The derivative of the integral is then given by: \[\frac{{dF(x)}}{{dx}} = f(b(x)) \cdot \frac{{db}}{{dx}} – f(a(x)) \cdot \frac{{da}}{{dx}}\]

Example: Differentiating an Integral

Let’s consider \(F(x) = \int_{{x^2}}^{{2x}} t^3 \, dt\). To find \(\frac{{dF}}{{dx}}\), we apply the chain rule: \[\frac{{dF(x)}}{{dx}} = (2x)^3 \cdot \frac{{d}}{{dx}}(2x) – (x^2)^3 \cdot \frac{{d}}{{dx}}(x^2)\]

Simplifying, we get: \[\frac{{dF(x)}}{{dx}} = 8x^3 – 3x^6\]

Conclusion

The Fundamental Theorem of Calculus allows us to leverage integral tables for both integration and differentiation. By applying the chain rule, we can find the derivatives of integrals with varying limits, making calculus computations more efficient and powerful.

Reading an Integral Table as a Derivative Table

An integral table can also be used as a derivative table. To do this, you need to understand the relationship between integrals and derivatives. Remember that the derivative of a function represents its rate of change, while the integral represents the accumulated area under the curve of the function.

When you encounter an integral table, you can read it backward to obtain derivatives. The table provides a list of common antiderivatives (indefinite integrals) and their corresponding derivatives. By looking up the antiderivative in the table, you can immediately deduce the derivative of the original function without having to go through the process of integration.

For example, suppose the integral table provides the following entry: \[\int \cos(x) \, dx = \sin(x) + C\]

Reading this entry backward, you can deduce the derivative of \(\sin(x)\) as: \[\frac{{d}}{{dx}}(\sin(x)) = \cos(x)\]

This method can save time and effort when finding derivatives of common functions. By leveraging the integral table, you can easily obtain derivatives for various functions, making it a valuable tool in calculus problem-solving.

Calculus Cheat Sheet

Applications of Derivatives

Tangent Line and Normal Line

The derivative of a function \(f(x)\) at a point \(x = a\) gives the slope of the tangent line to the graph of \(f\) at that point: \[f'(a) = \text{{Slope of tangent line}}\]

The slope of the normal line to the graph of \(f\) at a point \(x = a\) is the negative reciprocal of the slope of the tangent line: \[\text{{Slope of normal line}} = -\frac{1}{{f'(a)}}\]

Maxima and Minima

Critical points occur when the derivative \(f'(x)\) is equal to zero or does not exist. To find potential maxima and minima, solve \(f'(x) = 0\) for \(x\) and test each critical point.

First Derivative Test

If \(f'(x) > 0\) on an interval, \(f\) is increasing on that interval. If \(f'(x) < 0\), \(f\) is decreasing. A critical point that changes from increasing to decreasing (or vice versa) is a local maximum or minimum, respectively.

Second Derivative Test

If \(f”(x) > 0\) at a critical point \(x = c\), \(f\) has a local minimum at \(x = c\). If \(f”(x) < 0\), \(f\) has a local maximum. If \(f”(x) = 0\), the test is inconclusive.

Inflection Points

Inflection points are where the concavity of \(f\) changes. They occur when \(f”(x) = 0\) or does not exist. Test each potential inflection point by checking the sign of \(f”(x)\) on either side.

Related Rates

Related rates problems involve finding how two or more related variables are changing with respect to time. Set up equations and differentiate implicitly with respect to time (\(t\)). Solve for the desired rate by plugging in known values.

Optimization

Optimization problems involve finding maximum or minimum values of a function subject to given constraints. Identify the objective function and the constraint equation(s). Use the constraint(s) to express one variable in terms of another and substitute into the objective function. Find critical points and endpoints, and test them using the first or second derivative test.

Fundamental Theorem of Calculus in Accumulation

The Fundamental Theorem of Calculus also has an important application in accumulation problems. It states that if \(F(x)\) is the antiderivative of a function \(f(x)\), then the accumulation of \(f(x)\) from \(a\) to \(b\) can be found by evaluating \(F(b) – F(a)\): \[\int_{{a}}^{{b}} f(x) \, dx = F(b) – F(a)\]

This theorem allows us to calculate the net accumulation or total change of a quantity over a given interval. It is widely used in various real-life scenarios involving rates of change, such as distance traveled, total work done, and net change in a population over time.

Calculus Cheat Sheet

Parametric Equations and Calculus of Parametric Equations

Parametric Equations

Parametric equations represent a set of equations that describe the coordinates of a point as functions of one or more independent parameters (usually \(t\)). A parametric curve is traced by evaluating the \(x\) and \(y\) coordinates at different \(t\) values.

Derivatives of Parametric Equations

To find derivatives of parametric equations \(x = f(t)\) and \(y = g(t)\), differentiate each component with respect to \(t\): \[\frac{{dx}}{{dt}} = f'(t) \quad \text{and} \quad \frac{{dy}}{{dt}} = g'(t)\]

Tangent Line to a Parametric Curve

The equation of the tangent line to a parametric curve at a point \((x_0, y_0)\) can be found using the derivatives of \(x\) and \(y\) at that point: \[y – y_0 = \frac{{dy}}{{dt}}\Bigg|_{{t = t_0}} \cdot (x – x_0) = \frac{{g'(t_0)}}{{f'(t_0)}} \cdot (x – x_0)\]

Length of a Parametric Curve

The length of a parametric curve given by \(x = f(t)\) and \(y = g(t)\) from \(t = a\) to \(t = b\) is calculated using the arc length formula: \[L = \int_{{a}}^{{b}} \sqrt{{\left(\frac{{dx}}{{dt}}\right)^2 + \left(\frac{{dy}}{{dt}}\right)^2}} \, dt\]

Area Enclosed by a Parametric Curve

To find the area enclosed by a parametric curve given by \(x = f(t)\) and \(y = g(t)\) from \(t = a\) to \(t = b\), use the formula: \[A = \int_{{a}}^{{b}} y \cdot \frac{{dx}}{{dt}} \, dt\]

Polar Coordinates and Polar Equations

Polar coordinates represent points in a plane using a distance \(r\) from the origin and an angle \(\theta\) from the positive \(x\)-axis. Polar equations describe curves in polar coordinates, often written as \(r = f(\theta)\).

Derivatives of Polar Equations

To find derivatives of polar equations \(r = f(\theta)\), use the chain rule to differentiate with respect to \(\theta\): \[\frac{{dr}}{{d\theta}} = f'(\theta)\]

Area Enclosed by a Polar Curve

To find the area enclosed by a polar curve given by \(r = f(\theta)\) from \(\theta = \alpha\) to \(\theta = \beta\), use the formula: \[A = \frac{1}{2} \int_{{\alpha}}^{{\beta}} f(\theta)^2 \, d\theta\]

Calculus Cheat Sheet

Series and Sequences

Sequences

A sequence is an ordered list of numbers \(a_1, a_2, a_3, \ldots\) called terms. The limit of a sequence is the value it approaches as \(n\) becomes arbitrarily large. A sequence can be:

• Convergent: If it has a finite limit as \(n \to \infty\).
• Divergent: If it does not approach a finite limit.

Series

A series is the sum of the terms of a sequence: \(\sum_{n=1}^{\infty} a_n\). The series can be:

• Convergent: If the sequence of partial sums has a finite limit as \(n \to \infty\).
• Divergent: If the sequence of partial sums does not approach a finite limit.

Geometric Series

A geometric series is a series of the form \(\sum_{n=0}^{\infty} ar^n\), where \(a\) is the first term and \(r\) is the common ratio. The geometric series converges if \(|r| < 1\) and diverges otherwise.

Telescoping Series

A telescoping series is a series in which many terms cancel each other out, leaving only a finite number of terms. To evaluate a telescoping series, write out several terms and simplify the expression by canceling common terms.

Harmonic Series

The harmonic series is a divergent series given by \(\sum_{n=1}^{\infty} \frac{1}{n}\). As \(n\) goes to infinity, the sum approaches infinity. The harmonic series diverges.

Tests for Convergence

• Integral Test: Compares the series to an improper integral to determine convergence.
• Comparison Test: Compares the series to another series to determine convergence.
• Limit Comparison Test: Compares the series to another series using the limit of the quotient of their terms to determine convergence.
• Alternating Series Test: Determines convergence of alternating series with decreasing absolute values of terms.
• Ratio Test: Uses the ratio of consecutive terms to determine convergence or divergence.
• Root Test: Uses the root of the absolute value of terms to determine convergence or divergence.

Power Series

A power series is an infinite series of the form \(\sum_{n=0}^{\infty} a_n(x – c)^n\), where \(a_n\) are constants and \(c\) is the center of the series. Power series are used to represent functions as infinite polynomials.

Taylor Series

A Taylor series is a type of power series that represents a function \(f(x)\) as an infinite polynomial centered at \(c\). The Taylor series expansion of \(f(x)\) around \(x = c\) is given by: \[f(x) = \sum_{n=0}^{\infty} \frac{{f^{(n)}(c)}}{{n!}}(x – c)^n\]

Maclaurin Series

A Maclaurin series is a special case of the Taylor series where the center \(c\) is \(0\). The Maclaurin series expansion of \(f(x)\) is given by: \[f(x) = \sum_{n=0}^{\infty} \frac{{f^{(n)}(0)}}{{n!}}x^n\]

Calculus Cheat Sheet

Differential Equations (AP Calculus BC)

Separable Differential Equations

A separable ODE is one that can be written in the form: \[g(y) \frac{{dy}}{{dx}} = h(x)\] To solve a separable ODE, follow these steps:

1. Separate variables \(x\) and \(y\) on each side of the equation.
2. Integrate both sides with respect to \(x\) and add a constant of integration \(C\).
3. Solve for \(y\) in terms of \(x\) to find the general solution.

Linear Differential Equations

A linear ODE is one that can be written in the form: \[\frac{{dy}}{{dx}} + p(x)y = q(x)\] where \(p(x)\) and \(q(x)\) are functions of \(x\). To solve a linear ODE, follow these steps:

1. Find the integrating factor \(e^{\int p(x) \, dx}\).
2. Multiply both sides of the equation by the integrating factor.
3. Integrate both sides with respect to \(x\) and add a constant of integration \(C\).
4. Solve for \(y\) in terms of \(x\) to find the general solution.

Homogeneous Differential Equations

A homogeneous ODE is one that can be written in the form: \[\frac{{dy}}{{dx}} = f\left(\frac{{y}}{{x}}\right)\] To solve a homogeneous ODE, use the substitution \(v = \frac{{y}}{{x}}\) to convert it into a separable ODE. Then follow the steps for solving separable ODEs.

Applications of Differential Equations (AP Calculus BC)

Differential equations have various applications in real-world scenarios, and some are commonly covered in the AP Calculus BC exam. These applications include:

• Population Growth and Decay: Modeling population growth or decay with first-order ODEs using the logistic model: \[\frac{{dP}}{{dt}} = kP \left(1 – \frac{{P}}{{M}}\right)\] where \(P\) is the population, \(t\) is time, \(k\) is the growth/decay rate, and \(M\) is the carrying capacity.
• Newton’s Law of Cooling: Describing the cooling of an object based on the temperature difference between the object and its surroundings: \[\frac{{dT}}{{dt}} = -k(T – T_s)\] where \(T\) is the temperature of the object, \(T_s\) is the temperature of the surroundings, and \(k\) is the cooling rate constant.
• Damped Harmonic Motion: Analyzing the motion of a damped harmonic oscillator, often modeled using the damped harmonic oscillator equation: \[m\frac{{d^2x}}{{dt^2}} + c\frac{{dx}}{{dt}} + kx = 0\] where \(m\) is the mass of the object, \(c\) is the damping coefficient, and \(k\) is the spring constant.
• Logistic Growth: Modeling population growth with limiting factors using logistic differential equations: \[\frac{{dP}}{{dt}} = rP \left(1 – \frac{{P}}{{M}}\right)\] where \(P\) is the population, \(t\) is time, \(r\) is the growth rate, and \(M\) is the carrying capacity.
• RC Circuits: Describing the behavior of resistors and capacitors in electrical circuits: \[RC\frac{{dQ}}{{dt}} + Q = V(t)\] where \(Q\) is the charge on the capacitor, \(t\) is time, \(R\) is the resistance, \(C\) is the capacitance, and \(V(t)\) is the voltage source.

Direction Fields and Slope Fields

Direction fields and slope fields are graphical representations of first-order differential equations. They help visualize the behavior of solutions and can be used to sketch solutions without solving the equation explicitly. To draw a direction/slope field, evaluate the slope of the differential equation at various points in the \(xy\)-plane and draw short line segments with those slopes at each point.

Approximate Solutions: Euler’s Method

Euler’s method is a numerical technique to approximate solutions of first-order ODEs. It involves using tangent lines to approximate the behavior of the solution at discrete points. To use Euler’s method, follow these steps:

1. Choose a step size \(h\).
2. Start at the initial condition \((x_0, y_0)\).
3. Use the derivative at that point to find the slope of the tangent line.
4. Approximate the next point using \(x_{i+1} = x_i + h\) and \(y_{i+1} = y_i + h \cdot f(x_i, y_i)\), where \(f(x_i, y_i)\) is the derivative at \((x_i, y_i)\).
5. Repeat the process until the desired endpoint is reached.

Calculus BC Cheat Sheet

Parametric Equations and Vectors

Parametric Equations

Parametric equations are a way to represent curves in the plane using two separate functions of a third variable (usually \(t\)). For a curve in the plane described by the functions \(x(t)\) and \(y(t)\), the parametric equations are given as: \[x = x(t)\] \[y = y(t)\]

Finding Derivatives of Parametric Functions: The derivatives of \(x\) and \(y\) with respect to \(t\) are given by: \[\frac{{dx}}{{dt}}\] \[\frac{{dy}}{{dt}}\]

Finding \(dy/dx\) in terms of \(\frac{{dy}}{{dt}}\) and \(\frac{{dx}}{{dt}}\): \[\frac{{dy}}{{dx}} = \frac{{\frac{{dy}}{{dt}}}}{{\frac{{dx}}{{dt}}}}\]

Second Derivatives of Parametric Functions: The second derivatives of \(x\) and \(y\) with respect to \(t\) are given by: \[\frac{{d^2x}}{{dt^2}}\] \[\frac{{d^2y}}{{dt^2}}\]

Vector-Valued Functions

Vector-valued functions are functions that take a scalar input (usually \(t\)) and produce a vector output. For a vector-valued function \(\mathbf{r}(t) = \langle x(t), y(t), z(t) \rangle\), the components represent the position of a point in three-dimensional space.

Finding Derivatives of Vector-Valued Functions: The derivative of \(\mathbf{r}\) with respect to \(t\) is given by: \[\mathbf{r}'(t) = \langle x'(t), y'(t), z'(t) \rangle\]

Second Derivatives of Vector-Valued Functions: The second derivative of \(\mathbf{r}\) with respect to \(t\) is given by: \[\mathbf{r}”(t) = \langle x”(t), y”(t), z”(t) \rangle\]

Accumulation of Change

Calculating the Accumulation of Change in Length Over an Interval Using a Definite Integral: The arc length \(s\) of a parametric curve represented by \(\mathbf{r}(t)\) over the interval \([a, b]\) is given by the definite integral: \[s = \int_a^b \left|\left|\mathbf{r}'(t)\right|\right| \, dt\]

Polar Coordinates

Finding Derivatives of Functions Written in Polar Coordinates: To find the derivative \(dy/dx\) of a polar function \(r = f(\theta)\), use the following relationship: \[\frac{{dy}}{{dx}} = \frac{{\frac{{dr}}{{d\theta}} \cdot \sin \theta + r \cdot \frac{{d\theta}}{{dx}}}}{{\frac{{dr}}{{d\theta}} \cdot \cos \theta – r \cdot \frac{{d\theta}}{{dx}}}}\]

Finding the Area of Regions Bounded by Polar Curves: The area \(A\) of a region bounded by the polar curves \(r = f(\theta)\) and \(r = g(\theta)\) over the interval \([a, b]\) is given by the definite integral: \[A = \frac{1}{2} \int_a^b \left[ f(\theta)^2 – g(\theta)^2 \right] \, d\theta\]