University Calculus | Unit C4: Directional Derivatives and Tangent Planes

Sections 1 to 6: directional derivatives, the gradient, steepest ascent, tangent planes, linearization, differentials, gradients and level surfaces第 1 至 6 节：方向导数、梯度、最陡上升方向、切平面、线性化、微分、梯度与等值面CALC III

Directional Derivatives, Gradients, and Tangent Planes方向导数、梯度与切平面

Show all working. The direction vector in a directional derivative must be a unit vector before you compute the dot product. State the gradient explicitly at the given point before projecting.展示所有解题步骤。在计算方向导数的点积之前，方向向量必须先化为单位向量。在作投影之前，请明确写出给定点处的梯度。

Q1MEDIUM CORE computing the gradient and directional derivative计算梯度与方向导数 [8 marks]

Let $f(x,y)=x^{2}y+y^{3}$.

(a) Compute $\nabla f(x,y)$ and evaluate it at the point $P=(1,2)$.计算 $\nabla f(x,y)$ 并在点 $P=(1,2)$ 处求值。 [3]

(b) Find the directional derivative of $f$ at $P$ in the direction of $\mathbf{v}=\langle 3,4\rangle$. You must first normalise $\mathbf{v}$ to a unit vector $\mathbf{u}$.求 $f$ 在点 $P$ 处沿 $\mathbf{v}=\langle 3,4\rangle$ 方向的方向导数。须先将 $\mathbf{v}$ 化为单位向量 $\mathbf{u}$。 [3]

(c) State the direction in which $f$ increases most rapidly at $P$, and give the maximum rate of increase.写出 $f$ 在点 $P$ 处增长最快的方向，并给出最大增长率。 [2]

Q2MEDIUM CORE tangent plane to a graph surface $z=f(x,y)$图形曲面 $z=f(x,y)$ 的切平面 [8 marks]

Let $f(x,y)=3x^{2}-xy+y^{2}$.

(a) Find the partial derivatives $f_{x}$ and $f_{y}$, and evaluate them at the point $Q=(1,-1)$.求偏导数 $f_{x}$ 与 $f_{y}$，并在点 $Q=(1,-1)$ 处求值。 [3]

(b) Write the equation of the tangent plane to the surface $z=f(x,y)$ at the point $\bigl(1,-1,f(1,-1)\bigr)$.写出曲面 $z=f(x,y)$ 在点 $\bigl(1,-1,f(1,-1)\bigr)$ 处的切平面方程。 [3]

Q3MEDIUM CORE linearization and the total differential线性化与全微分 [6 marks]

Let $f(x,y)=\sqrt{x^{2}+y^{2}}$. Let $P_{0}=(3,4)$.

(a) Write the linearization $L(x,y)$ of $f$ at $P_{0}$.写出 $f$ 在 $P_{0}$ 处的线性化 $L(x,y)$。 [3]

(b) Write the total differential $df$ at $P_{0}$, and use it to estimate the change in $f$ when $(x,y)$ moves from $(3,4)$ to $(3.06, 3.92)$.写出 $f$ 在 $P_{0}$ 处的全微分 $df$，并用它估计当 $(x,y)$ 从 $(3,4)$ 移动到 $(3.06, 3.92)$ 时 $f$ 的变化量。 [3]

Q4HARD CORE direction of zero change along a level curve沿等值线方向变化率为零的方向 [6 marks]

Let $g(x,y)=x^{2}+4y^{2}$ and let $P=(2,1)$.

(a) Compute $\nabla g(2,1)$ and the maximum rate of change of $g$ at $P$.计算 $\nabla g(2,1)$ 及 $g$ 在点 $P$ 处的最大变化率。 [2]

(b) Find a unit vector $\mathbf{u}$ in which $D_{\mathbf{u}}g(2,1)=0$. Explain geometrically why such a direction exists and what it represents.找一个单位向量 $\mathbf{u}$，使得 $D_{\mathbf{u}}g(2,1)=0$。从几何角度说明为何存在这样的方向以及它的含义。 [4]

Proofs About the Gradient and Directional Derivative梯度与方向导数的证明

These items are graded on the logic of the argument, not just the final answer. Cite the theorem or inequality you use (dot-product formula, Cauchy-Schwarz) and verify every hypothesis before invoking it.本部分按论证逻辑评分，而不仅看最终答案。引用所用定理或不等式（点积公式、柯西-施瓦茨不等式），并在使用前验证每个前提条件。

Q5HARD PROOF deriving $D_{\mathbf{u}}f = \nabla f \cdot \mathbf{u}$ from the limit definition从极限定义推导 $D_{\mathbf{u}}f = \nabla f \cdot \mathbf{u}$ [10 marks]

Let $f:\mathbb{R}^{2}\to\mathbb{R}$ be differentiable at a point $\mathbf{a}=(a_{1},a_{2})$, and let $\mathbf{u}=(u_{1},u_{2})$ be a unit vector.

(a) Write the limit-definition formula for $D_{\mathbf{u}}f(\mathbf{a})$. Then write the first-order Taylor (differentiability) condition for $f$ at $\mathbf{a}$: that is, expand $f(\mathbf{a}+h\mathbf{u})-f(\mathbf{a})$ using the definition of differentiability, expressing the result in terms of $f_{x}(\mathbf{a})$, $f_{y}(\mathbf{a})$, $h$, $u_{1}$, $u_{2}$, and an error term $\varepsilon(h)$ satisfying $\varepsilon(h)/h\to 0$.写出 $D_{\mathbf{u}}f(\mathbf{a})$ 的极限定义公式。然后写出 $f$ 在 $\mathbf{a}$ 处的一阶泰勒（可微性）条件：即利用可微性定义展开 $f(\mathbf{a}+h\mathbf{u})-f(\mathbf{a})$，结果用 $f_{x}(\mathbf{a})$、$f_{y}(\mathbf{a})$、$h$、$u_{1}$、$u_{2}$ 以及满足 $\varepsilon(h)/h\to 0$ 的误差项 $\varepsilon(h)$ 表示。 [4]

(b) Divide by $h$ and take $h\to 0$ to conclude that $D_{\mathbf{u}}f(\mathbf{a})=\nabla f(\mathbf{a})\cdot\mathbf{u}$. State clearly why the error term vanishes.两边除以 $h$ 并令 $h\to 0$，从而得出 $D_{\mathbf{u}}f(\mathbf{a})=\nabla f(\mathbf{a})\cdot\mathbf{u}$。清楚说明误差项消失的原因。 [3]

(c) Use the Cauchy-Schwarz inequality $|\mathbf{a}\cdot\mathbf{b}|\le|\mathbf{a}||\mathbf{b}|$ to show that $D_{\mathbf{u}}f(\mathbf{a})\le|\nabla f(\mathbf{a})|$ for every unit vector $\mathbf{u}$, with equality when $\mathbf{u}=\nabla f(\mathbf{a})/|\nabla f(\mathbf{a})|$.利用柯西-施瓦茨不等式 $|\mathbf{a}\cdot\mathbf{b}|\le|\mathbf{a}||\mathbf{b}|$ 证明对所有单位向量 $\mathbf{u}$ 均有 $D_{\mathbf{u}}f(\mathbf{a})\le|\nabla f(\mathbf{a})|$，且当 $\mathbf{u}=\nabla f(\mathbf{a})/|\nabla f(\mathbf{a})|$ 时等号成立。 [3]

Q6HARD PROOF gradient is normal to a level surface梯度垂直于等值面 [8 marks]

Let $F:\mathbb{R}^{3}\to\mathbb{R}$ be differentiable and let $S$ be the level surface $F(x,y,z)=k$. Let $\mathbf{r}(t)$ be any smooth curve lying on $S$ with $\mathbf{r}(0)=\mathbf{p}_{0}$ and $\mathbf{r}'(0)=\mathbf{T}$.

(a) Since $\mathbf{r}(t)$ lies on $S$, we have $F(\mathbf{r}(t))=k$ for all $t$. Differentiate both sides with respect to $t$ using the chain rule and evaluate at $t=0$ to obtain a relation between $\nabla F(\mathbf{p}_{0})$ and $\mathbf{T}$.由于 $\mathbf{r}(t)$ 在 $S$ 上，对所有 $t$ 有 $F(\mathbf{r}(t))=k$。用链式法则对两边关于 $t$ 求导，并在 $t=0$ 处求值，得出 $\nabla F(\mathbf{p}_{0})$ 与 $\mathbf{T}$ 之间的关系。 [4]

(b) Explain why the relation you found in (a) proves that $\nabla F(\mathbf{p}_{0})$ is orthogonal to every tangent vector of $S$ at $\mathbf{p}_{0}$, and hence is a normal vector to $S$ there.解释为何 (a) 中所得关系证明了 $\nabla F(\mathbf{p}_{0})$ 与 $S$ 在 $\mathbf{p}_{0}$ 处的每一切向量都正交，因而是 $S$ 在该点的法向量。 [2]

(c) Write the equation of the tangent plane to $S$ at $\mathbf{p}_{0}=(x_{0},y_{0},z_{0})$ in terms of $F_{x}$, $F_{y}$, $F_{z}$ evaluated at $\mathbf{p}_{0}$.用在 $\mathbf{p}_{0}$ 处计算的 $F_{x}$、$F_{y}$、$F_{z}$ 写出 $S$ 在 $\mathbf{p}_{0}=(x_{0},y_{0},z_{0})$ 处的切平面方程。 [2]

Q7HARD PROOF tangent plane to an implicit surface; normal line隐式曲面的切平面与法线 [8 marks]

Consider the ellipsoid $F(x,y,z)=x^{2}+2y^{2}+3z^{2}=6$ and the point $P_{0}=(1,1,1)$.

(a) Verify that $P_{0}$ lies on the ellipsoid. Then compute $\nabla F(1,1,1)$.验证 $P_{0}$ 在椭球面上，然后计算 $\nabla F(1,1,1)$。 [2]

(b) Write the equation of the tangent plane to the ellipsoid at $P_{0}$.写出椭球面在 $P_{0}$ 处的切平面方程。 [3]

(c) Write parametric equations for the normal line to the ellipsoid at $P_{0}$. Then find the point where the normal line intersects the $xy$-plane (set $z=0$).写出椭球面在 $P_{0}$ 处法线的参数方程，然后求法线与 $xy$ 平面（令 $z=0$）的交点。 [3]

Steepest Ascent, Linearization Error, and Synthesis最陡上升、线性化误差与综合

Set up each problem cleanly. Carry exact values through all intermediate steps. State any approximation explicitly, and bound your error where asked.清晰建立每道题的模型。在所有中间步骤中保留精确值。明确写出任何近似，并在要求处给出误差界。

Q8HARD APPLIED steepest-ascent path in a temperature field温度场中的最陡上升路径 [10 marks]

The temperature at a point $(x,y)$ on a metal plate is $T(x,y)=100-2x^{2}-3y^{2}$ (degrees Celsius). A heat-seeking particle is placed at the point $A=(3,2)$.金属板上点 $(x,y)$ 处的温度为 $T(x,y)=100-2x^{2}-3y^{2}$（摄氏度）。一个趋热粒子被放置在点 $A=(3,2)$。

(a) Compute $\nabla T(3,2)$. In which direction (give a unit vector) should the particle move to ascend the temperature most steeply? What is the maximum rate of temperature increase per unit distance at $A$?计算 $\nabla T(3,2)$。粒子应沿哪个方向（给出单位向量）移动才能使温度上升最陡？在点 $A$ 处每单位距离的最大温度增长率是多少？ [4]

(b) What is the directional derivative of $T$ at $A$ in the direction toward the origin? (You must normalise the direction vector.) What does the sign of your answer mean physically?$T$ 在点 $A$ 处沿指向原点方向的方向导数是多少？（须将方向向量化为单位向量。）答案的符号在物理上意味着什么？ [3]

(c) The particle moves in the steepest-ascent direction for a small step of length $\Delta s = 0.1$. Estimate the temperature change using the linearization, and state the temperature at $A$ before and after the step (to two decimal places).粒子沿最陡上升方向移动步长 $\Delta s = 0.1$。用线性化估计温度变化量，并写出步骤前后点 $A$ 处的温度（保留两位小数）。 [3]

Q9HARD APPLIED linearization error bound for a computed value计算值的线性化误差界 [10 marks]

An engineer measures the legs of a right triangle as $a=6.00$ cm and $b=8.00$ cm, each with a possible measurement error of at most $\pm 0.05$ cm. The hypotenuse is computed as $c=\sqrt{a^{2}+b^{2}}$.工程师测量一个直角三角形的两直角边分别为 $a=6.00$ cm 和 $b=8.00$ cm，每条边的测量误差不超过 $\pm 0.05$ cm。斜边由 $c=\sqrt{a^{2}+b^{2}}$ 计算得出。

(a) Compute the exact value of $c$ at $(a,b)=(6,8)$ and write the linearization $L(a,b)$ of $c=\sqrt{a^{2}+b^{2}}$ at the point $(6,8)$.计算 $(a,b)=(6,8)$ 时 $c$ 的精确值，并写出 $c=\sqrt{a^{2}+b^{2}}$ 在点 $(6,8)$ 处的线性化 $L(a,b)$。 [3]

(b) Write the total differential $dc$ at $(6,8)$ in terms of $da$ and $db$. Use it to find the maximum absolute error $|dc|_{\max}$ in the computed hypotenuse, given $|da|\le 0.05$ and $|db|\le 0.05$.用 $da$ 和 $db$ 写出 $(6,8)$ 处的全微分 $dc$。利用它求斜边计算值的最大绝对误差 $|dc|_{\max}$，已知 $|da|\le 0.05$ 且 $|db|\le 0.05$。 [4]

(c) Find the maximum relative error $|dc|_{\max}/c$ and express it as a percentage. Identify which of $a$ or $b$ contributes more to the absolute error in $c$, and explain why.求最大相对误差 $|dc|_{\max}/c$ 并以百分比表示。判断 $a$ 与 $b$ 中哪一个对 $c$ 的绝对误差贡献更大，并说明原因。 [3]

Q10HARD APPLIED synthesis: gradient, tangent plane, and directional derivative on one surface综合：同一曲面上的梯度、切平面与方向导数 [8 marks]

Consider the surface $z=f(x,y)=xe^{y-1}-y^{2}+1$ and the point $P=(1,1,1)$. (Verify $P$ lies on the surface before starting.)考虑曲面 $z=f(x,y)=xe^{y-1}-y^{2}+1$ 及点 $P=(1,1,1)$。（开始前验证 $P$ 在曲面上。）

(a) Compute $\nabla f(1,1)$ and write the equation of the tangent plane to the surface at $P$.计算 $\nabla f(1,1)$ 并写出曲面在点 $P$ 处的切平面方程。 [3]

(b) Find the directional derivative of $f$ at $(1,1)$ in the direction of steepest descent, and state the maximum rate of decrease.求 $f$ 在点 $(1,1)$ 处沿最陡下降方向的方向导数，并写出最大下降率。 [2]

(c) The surface can also be written as the level surface $F(x,y,z)=xe^{y-1}-y^{2}+1-z=0$. Confirm that the normal vector $\nabla F(1,1,1)$ is consistent with the tangent plane you found in (a).该曲面也可写成等值面 $F(x,y,z)=xe^{y-1}-y^{2}+1-z=0$。验证法向量 $\nabla F(1,1,1)$ 与 (a) 中所求切平面相符。 [3]

Directional Derivatives and Tangent Planes方向导数与切平面

Directional Derivatives, Gradients, and Tangent Planes方向导数、梯度与切平面

Proofs About the Gradient and Directional Derivative梯度与方向导数的证明

Steepest Ascent, Linearization Error, and Synthesis最陡上升、线性化误差与综合