Calculus Reference

Cheatsheet Content

### CA1: Functions of Several Variables #### Definition & Domain - A function $f(x, y)$ assigns a value to each input pair $(x, y)$ in its **domain**. - The domain is the set of all possible input values for which the function is defined. - **Example**: For $f(x,y) = \sqrt{x-y^2}$, the domain requires $x-y^2 \ge 0$, so $x \ge y^2$. This is the region to the right of or on the parabola $x=y^2$. #### Contour Plots (Level Curves) - **Definition**: A contour plot shows curves where $f(x, y) = k$ for various constant values of $k$. These are also called level curves. - **Interpretation**: - Closely spaced contour lines indicate a steep slope. - Widely spaced contour lines indicate a gentle slope. - Concentric circles/ellipses often indicate local maxima or minima. - **Example**: For $f(x,y) = x^2+y^2$, level curves are circles $x^2+y^2=k$ (for $k>0$). #### Surfaces of Revolution - To rotate a curve given by $z = f(x)$, $x \ge 0$ (in the $xz$-plane) about the $z$-axis, replace $x$ with $\sqrt{x^2+y^2}$. - To rotate a curve given by $z = f(y)$, $y \ge 0$ (in the $yz$-plane) about the $z$-axis, replace $y$ with $\sqrt{x^2+y^2}$. - **Example**: Rotating $z = e^{-x^2}$ ($x \ge 0$) about the $z$-axis yields $z = e^{-(x^2+y^2)}$. #### Limits and Continuity - **Limit Definition**: $\lim_{(x,y) \to (a,b)} f(x,y) = L$ if for every $\epsilon > 0$ there exists a $\delta > 0$ such that if $0 ### CA2: Partial Differentiation #### Definition and Notation - For a function $f(x,y,z,...)$, a partial derivative is the derivative with respect to one variable, treating all other variables as constants. - **First-Order**: - With respect to $x$: $f_x = \frac{\partial f}{\partial x}$ (treat $y, z, ...$ as constants) - With respect to $y$: $f_y = \frac{\partial f}{\partial y}$ (treat $x, z, ...$ as constants) - **Example**: Let $f(x,y,z) = x^2y - y^3z + \frac{x}{z}$. - $f_x = 2xy + \frac{1}{z}$ - $f_y = x^2 - 3y^2z$ - $f_z = -y^3 - \frac{x}{z^2}$ #### Higher-Order Partial Derivatives - **Second-order**: - $f_{xx} = \frac{\partial^2 f}{\partial x^2} = \frac{\partial}{\partial x}\left(\frac{\partial f}{\partial x}\right)$ - $f_{yy} = \frac{\partial^2 f}{\partial y^2} = \frac{\partial}{\partial y}\left(\frac{\partial f}{\partial y}\right)$ - $f_{zz} = \frac{\partial^2 f}{\partial z^2} = \frac{\partial}{\partial z}\left(\frac{\partial f}{\partial z}\right)$ - **Mixed partials**: - $f_{xy} = \frac{\partial^2 f}{\partial y \partial x} = \frac{\partial}{\partial y}\left(\frac{\partial f}{\partial x}\right)$ - $f_{yx} = \frac{\partial^2 f}{\partial x \partial y} = \frac{\partial}{\partial x}\left(\frac{\partial f}{\partial y}\right)$ #### Clairaut's Theorem (Equality of Mixed Partials) - If the mixed partial derivatives $f_{xy}$ and $f_{yx}$ are continuous on some open disk, then $f_{xy} = f_{yx}$ on that disk. - This theorem simplifies calculations as the order of differentiation often doesn't matter. #### The Laplacian Operator - For $f(x,y)$: $\nabla^2 f = f_{xx} + f_{yy}$ - For $f(x,y,z)$: $\nabla^2 f = f_{xx} + f_{yy} + f_{zz}$ - If $\nabla^2 f = 0$, $f$ is called a harmonic function. ### CA3: Linear Approximations & Tangent Planes #### Linear Approximation (for $f(x,y)$) - The linear approximation (or linearization) of $f(x,y)$ at a point $(x_0, y_0)$ is given by: $$L(x,y) = f(x_0,y_0) + f_x(x_0,y_0)(x-x_0) + f_y(x_0,y_0)(y-y_0)$$ - This approximation is good for points $(x,y)$ near $(x_0,y_0)$. It represents the function's value on the tangent plane at $(x_0,y_0)$. #### Tangent Plane Equation - The equation of the tangent plane to the surface $z = f(x,y)$ at the point $(x_0, y_0, z_0)$ where $z_0 = f(x_0,y_0)$ is: $$z - z_0 = f_x(x_0,y_0)(x-x_0) + f_y(x_0,y_0)(y-y_0)$$ - For implicitly defined surfaces $F(x,y,z)=0$, the tangent plane at $(x_0,y_0,z_0)$ is: $$F_x(x_0,y_0,z_0)(x-x_0) + F_y(x_0,y_0,z_0)(y-y_0) + F_z(x_0,y_0,z_0)(z-z_0) = 0$$ #### Example: Tangent Plane Find the equation for the tangent plane to $z = x^2 + 2y^2$ at the point $(1, 1, 3)$. 1. **Function and Point**: $f(x,y) = x^2 + 2y^2$, $(x_0,y_0) = (1,1)$, $z_0 = 3$. 2. **Partial Derivatives**: - $f_x = 2x$ - $f_y = 4y$ 3. **Evaluate at Point**: - $f_x(1,1) = 2(1) = 2$ - $f_y(1,1) = 4(1) = 4$ 4. **Tangent Plane Equation**: - $z - 3 = 2(x-1) + 4(y-1)$ - $z - 3 = 2x - 2 + 4y - 4$ - $z = 2x + 4y - 3$ #### Differentials and Error Estimation - **Total Differential**: For $z = f(x,y)$, the total differential $dz$ is: $$dz = \frac{\partial f}{\partial x}dx + \frac{\partial f}{\partial y}dy$$ where $dx = \Delta x$ and $dy = \Delta y$ are small changes in $x$ and $y$. - **Error Estimation**: The maximum possible error $\Delta f$ in calculating $f(x_1, ..., x_n)$ given errors $\Delta x_i$ in each variable is approximately: $$\Delta f \approx \left|\frac{\partial f}{\partial x_1}\Delta x_1\right| + \left|\frac{\partial f}{\partial x_2}\Delta x_2\right| + \dots + \left|\frac{\partial f}{\partial x_n}\Delta x_n\right|$$ - **Example**: The volume of a cylindrical can is $V = \pi r^2 h$. If $r=4$ cm and $h=10$ cm, and $dr = \pm 0.1$ cm, $dh = \pm 0.2$ cm, estimate the maximum error in $V$. 1. $V_r = 2\pi rh = 2\pi(4)(10) = 80\pi$ 2. $V_h = \pi r^2 = \pi(4^2) = 16\pi$ 3. $\Delta V \approx |V_r dr| + |V_h dh| = |80\pi (\pm 0.1)| + |16\pi (\pm 0.2)| = 8\pi + 3.2\pi = 11.2\pi$ cm$^3$. ### CA4: Directional Derivatives & Gradient Vectors #### Gradient Vector - For a function $f(x,y)$, the gradient vector is: $$\nabla f(x,y) = \left\langle \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right\rangle = f_x \mathbf{i} + f_y \mathbf{j}$$ - For $f(x,y,z)$: $$\nabla f(x,y,z) = \left\langle \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right\rangle = f_x \mathbf{i} + f_y \mathbf{j} + f_z \mathbf{k}$$ - **Properties of the Gradient**: 1. $\nabla f$ points in the direction of the greatest rate of increase of $f$. 2. Its magnitude $|\nabla f|$ is the maximum rate of increase. 3. $\nabla f$ is orthogonal (perpendicular) to the level curves of $f(x,y)$ or level surfaces of $f(x,y,z)$. #### Directional Derivative - The directional derivative of $f$ at $(x_0,y_0)$ (or $(x_0,y_0,z_0)$) in the direction of a unit vector $\mathbf{u}$ is: $$D_{\mathbf{u}}f(x_0,y_0) = \nabla f(x_0,y_0) \cdot \mathbf{u}$$ where $\mathbf{u} = \frac{\mathbf{v}}{|\mathbf{v}|}$ is the unit vector in the direction of $\mathbf{v}$. #### Example: Direction and Rate of Max Increase Find the direction of maximum increase of $f(x,y) = x^2e^{3y}$ at $(2,0)$ and the maximum rate of increase. 1. **Partial Derivatives**: - $f_x = 2xe^{3y}$ - $f_y = 3x^2e^{3y}$ 2. **Gradient Vector**: $\nabla f = \langle 2xe^{3y}, 3x^2e^{3y} \rangle$. 3. **Evaluate Gradient at Point**: At $(2,0)$: - $\nabla f(2,0) = \langle 2(2)e^{3(0)}, 3(2)^2e^{3(0)} \rangle = \langle 4, 12 \rangle$. 4. **Direction of Max Increase**: $\mathbf{u} = \nabla f(2,0) = \langle 4, 12 \rangle$. 5. **Maximum Rate of Increase**: $|\nabla f(2,0)| = \sqrt{4^2 + 12^2} = \sqrt{16 + 144} = \sqrt{160} = 4\sqrt{10}$. #### Example: Directional Derivative Find the directional derivative of $f(x,y,z) = xyz$ at $(1,2,3)$ in the direction of $\mathbf{v} = \langle 1,1,-1 \rangle$. 1. **Partial Derivatives**: $f_x = yz$, $f_y = xz$, $f_z = xy$. 2. **Gradient Vector**: $\nabla f = \langle yz, xz, xy \rangle$. 3. **Evaluate Gradient at Point**: At $(1,2,3)$: - $\nabla f(1,2,3) = \langle (2)(3), (1)(3), (1)(2) \rangle = \langle 6, 3, 2 \rangle$. 4. **Unit Vector**: - $|\mathbf{v}| = \sqrt{1^2+1^2+(-1)^2} = \sqrt{3}$. - $\mathbf{u} = \frac{1}{\sqrt{3}}\langle 1,1,-1 \rangle = \left\langle \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, -\frac{1}{\sqrt{3}} \right\rangle$. 5. **Directional Derivative**: - $D_{\mathbf{u}}f(1,2,3) = \nabla f(1,2,3) \cdot \mathbf{u} = \langle 6, 3, 2 \rangle \cdot \left\langle \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, -\frac{1}{\sqrt{3}} \right\rangle$ - $= \frac{6}{\sqrt{3}} + \frac{3}{\sqrt{3}} - \frac{2}{\sqrt{3}} = \frac{7}{\sqrt{3}}$. ### CA5: Local and Global Extrema #### Critical Points - A critical point $(x_0, y_0)$ of $f(x,y)$ is a point where: 1. $\nabla f(x_0, y_0) = \langle 0, 0 \rangle$ (i.e., $f_x(x_0,y_0) = 0$ and $f_y(x_0,y_0) = 0$), OR 2. Either $f_x(x_0,y_0)$ or $f_y(x_0,y_0)$ does not exist. #### Second Derivative Test (for Local Extrema) - Let $(x_0, y_0)$ be a critical point of $f(x,y)$ and define the discriminant $D$: $$D = D(x_0,y_0) = f_{xx}(x_0,y_0)f_{yy}(x_0,y_0) - [f_{xy}(x_0,y_0)]^2$$ 1. If $D > 0$ and $f_{xx}(x_0,y_0) > 0$, then $f$ has a **local minimum** at $(x_0,y_0)$. 2. If $D > 0$ and $f_{xx}(x_0,y_0) 0$ and $f_{xx} = 6 > 0$, $(1,1)$ is a **local minimum**. #### Global Extrema on a Closed Bounded Region To find the absolute maximum and minimum values of $f(x,y)$ on a closed bounded region $D$: 1. Find the values of $f$ at the critical points of $f$ that lie *inside* $D$. 2. Find the extreme values of $f$ on the boundary of $D$. This often involves parameterizing the boundary segments and reducing the problem to a single-variable optimization problem for each segment. 3. Compare all the values found in steps 1 and 2. The largest value is the absolute maximum, and the smallest is the absolute minimum. #### Example: Global Extrema Find the absolute maximum and minimum of $f(x,y) = xy$ on the square region $D = \{(x,y) \mid 0 \le x \le 2, \ 0 \le y \le 2\}$. 1. **Critical Points in D**: - $f_x = y = 0$ - $f_y = x = 0$ - Critical point $(0,0)$. $f(0,0)=0$. (This point is on the boundary). 2. **Boundary of D**: - **Segment 1**: $y=0$, $0 \le x \le 2$. $f(x,0) = x(0) = 0$. Min/Max on this segment is 0. - **Segment 2**: $x=2$, $0 \le y \le 2$. $f(2,y) = 2y$. $g(y)=2y$. On $[0,2]$, min is $g(0)=0$, max is $g(2)=4$. - **Segment 3**: $y=2$, $0 \le x \le 2$. $f(x,2) = 2x$. $h(x)=2x$. On $[0,2]$, min is $h(0)=0$, max is $h(2)=4$. - **Segment 4**: $x=0$, $0 \le y \le 2$. $f(0,y) = 0y = 0$. Min/Max on this segment is 0. 3. **Compare Values**: $\{0, 4\}$. - Absolute Maximum: $4$ (at $(2,2)$). - Absolute Minimum: $0$ (along the axes). ### CA6: Chain Rule & Implicit Differentiation #### Chain Rule for Functions of Several Variables 1. **Case 1: $z = f(x,y)$, where $x = g(t)$ and $y = h(t)$** $$\frac{dz}{dt} = \frac{\partial z}{\partial x}\frac{dx}{dt} + \frac{\partial z}{\partial y}\frac{dy}{dt}$$ 2. **Case 2: $z = f(x,y)$, where $x = g(s,t)$ and $y = h(s,t)$** $$\frac{\partial z}{\partial s} = \frac{\partial z}{\partial x}\frac{\partial x}{\partial s} + \frac{\partial z}{\partial y}\frac{\partial y}{\partial s}$$ $$\frac{\partial z}{\partial t} = \frac{\partial z}{\partial x}\frac{\partial x}{\partial t} + \frac{\partial z}{\partial y}\frac{\partial y}{\partial t}$$ 3. **General Case**: If $u = f(x_1, ..., x_n)$ and each $x_i = g_i(t_1, ..., t_m)$, then $$\frac{\partial u}{\partial t_j} = \sum_{i=1}^{n} \frac{\partial u}{\partial x_i}\frac{\partial x_i}{\partial t_j}$$ #### Example: Chain Rule Let $z = x^2y + y^2$, where $x = \sin t$ and $y = e^t$. Find $\frac{dz}{dt}$. 1. **Partial Derivatives of $z$**: - $\frac{\partial z}{\partial x} = 2xy$ - $\frac{\partial z}{\partial y} = x^2 + 2y$ 2. **Derivatives of $x, y$ with respect to $t$**: - $\frac{dx}{dt} = \cos t$ - $\frac{dy}{dt} = e^t$ 3. **Apply Chain Rule**: - $\frac{dz}{dt} = (2xy)(\cos t) + (x^2 + 2y)(e^t)$ - Substitute $x=\sin t, y=e^t$: - $\frac{dz}{dt} = (2(\sin t)(e^t))(\cos t) + ((\sin t)^2 + 2e^t)(e^t)$ - $\frac{dz}{dt} = 2e^t \sin t \cos t + e^t \sin^2 t + 2e^{2t}$. #### Implicit Differentiation - If an equation $F(x,y) = 0$ defines $y$ implicitly as a differentiable function of $x$, then: $$\frac{dy}{dx} = -\frac{F_x}{F_y}$$ - If an equation $F(x,y,z) = 0$ defines $z$ implicitly as a differentiable function of $x$ and $y$, then: $$\frac{\partial z}{\partial x} = -\frac{F_x}{F_z}$$ $$\frac{\partial z}{\partial y} = -\frac{F_y}{F_z}$$ #### Example: Implicit Differentiation Find $\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y}$ if $x^2y + y^2z + z^2x = 10$. 1. **Define F**: $F(x,y,z) = x^2y + y^2z + z^2x - 10 = 0$. 2. **Partial Derivatives of F**: - $F_x = 2xy + z^2$ - $F_y = x^2 + 2yz$ - $F_z = y^2 + 2zx$ 3. **Apply Formulas**: - $\frac{\partial z}{\partial x} = -\frac{F_x}{F_z} = -\frac{2xy + z^2}{y^2 + 2zx}$ - $\frac{\partial z}{\partial y} = -\frac{F_y}{F_z} = -\frac{x^2 + 2yz}{y^2 + 2zx}$ ### CA7: Maclaurin Series ($a=0$) #### Definition A Maclaurin series is a Taylor series expansion of a function about $x=0$. The Maclaurin series for $f(x)$ is given by: $$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}x^n = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \dots$$ #### Common Maclaurin Series Expansions - **$e^x$**: $\sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots$ (Converges for all $x \in (-\infty, \infty)$) - **$\sin x$**: $\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!}x^{2n+1} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \dots$ (Converges for all $x \in (-\infty, \infty)$) - **$\cos x$**: $\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!}x^{2n} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \dots$ (Converges for all $x \in (-\infty, \infty)$) - **$\frac{1}{1-x}$ (Geometric Series)**: $\sum_{n=0}^{\infty} x^n = 1 + x + x^2 + x^3 + \dots$ (Converges for $|x| ### CA8: Taylor Series (centered at $x=a$) #### Definition The Taylor series of a function $f(x)$ centered at $x=a$ is given by: $$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \dots$$ A Maclaurin series is a special case of a Taylor series where $a=0$. #### Radius and Interval of Convergence - **Ratio Test**: For a series $\sum a_n$, compute $L = \lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right|$. - If $L 1$, the series diverges. - If $L = 1$, the test is inconclusive (check endpoints separately). - The **radius of convergence** $R$ is found by solving $L