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Tangent planes practice problems

The plane that touches a surface z = f(x, y) at a point. 25 problems with worked solutions; in 25 of them every equation is proved by a computer algebra system.

3 worked problems

Try each one before opening the solution. Every line is labeled with what checked it. How we verify

Problem 10.88

hard
Find the tangent plane to \( \displaystyle z = e^{x - y} \) at the point \( \displaystyle (1, 2, e^{-1}) \).
  1. z = f(a,b) + f_x(a,b)(x − a) + f_y(a,b)(y − b).
    Reviewed
  2. \[ \left. \frac{\partial}{\partial x} e^{x - y} \right|_{\substack{ x=1\\ y=2 }} = e^{-1} \]
    f_x at the point.✓ Proved
  3. \[ \left. \frac{\partial}{\partial y} e^{x - y} \right|_{\substack{ x=1\\ y=2 }} = - \frac{1}{e} \]
    f_y at the point.✓ Proved
  4. \[ \frac{x - 1}{e} - \frac{y - 2}{e} + e^{-1} = \frac{x}{e} - \frac{y}{e} + \frac{2}{e} \]
    The plane.✓ Proved
Answer \( z = \frac{x}{e} - \frac{y}{e} + \frac{2}{e} \)

✓ Nihil obstat Lines: 3 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 10.87

medium
Find the tangent plane to \( \displaystyle z = e^{x - y} \) at the point \( \displaystyle (0, 0, 1) \).
  1. z = f(a,b) + f_x(a,b)(x − a) + f_y(a,b)(y − b).
    Reviewed
  2. \[ \left. \frac{\partial}{\partial x} e^{x - y} \right|_{\substack{ x=0\\ y=0 }} = 1 \]
    f_x at the point.✓ Proved
  3. \[ \left. \frac{\partial}{\partial y} e^{x - y} \right|_{\substack{ x=0\\ y=0 }} = -1 \]
    f_y at the point.✓ Proved
  4. \[ x - y + 1 \]
    The plane.Reviewed
Answer \( z = x - y + 1 \)

✓ Nihil obstat Lines: 2 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 10.91

medium
Find the tangent plane to \( \displaystyle z = e^{x - y} \) at the point \( \displaystyle (2, 1, e) \).
  1. z = f(a,b) + f_x(a,b)(x − a) + f_y(a,b)(y − b).
    Reviewed
  2. \[ \left. \frac{\partial}{\partial x} e^{x - y} \right|_{\substack{ x=2\\ y=1 }} = e \]
    f_x at the point.✓ Proved
  3. \[ \left. \frac{\partial}{\partial y} e^{x - y} \right|_{\substack{ x=2\\ y=1 }} = - e \]
    f_y at the point.✓ Proved
  4. \[ e \left(x - 2\right) - e \left(y - 1\right) + e = e x - e y \]
    The plane.✓ Proved
Answer \( z = e x - e y \)

✓ Nihil obstat Lines: 3 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

More tangent planes practice

Find the tangent plane to \( \displaystyle z = x^{2} y + 1 \) at the point \( \displaystyle (1, 1, 2) \).
Problem 10.24medium✓ Every equation proved
Find the tangent plane to \( \displaystyle z = x^{2} y + 1 \) at the point \( \displaystyle (1, 2, 3) \).
Problem 10.25medium✓ Every equation proved
Find the tangent plane to \( \displaystyle z = x^{2} + y^{2} \) at the point \( \displaystyle (0, 0, 0) \).
Problem 10.26medium✓ Every equation proved
Find the tangent plane to \( \displaystyle z = x^{2} + y^{2} \) at the point \( \displaystyle (1, -1, 2) \).
Problem 10.27medium✓ Every equation proved
Find the tangent plane to \( \displaystyle z = x y \) at the point \( \displaystyle (1, 1, 1) \).
Problem 10.28medium✓ Every equation proved
Find the tangent plane to \( \displaystyle z = x^{2} + y^{2} \) at the point \( \displaystyle (1, 2, 5) \).
Problem 10.29medium✓ Every equation proved
Find the tangent plane to \( \displaystyle z = x^{2} y + 1 \) at the point \( \displaystyle (1, -1, 0) \).
Problem 10.30medium✓ Every equation proved
Find the tangent plane to \( \displaystyle z = x^{2} - y^{2} \) at the point \( \displaystyle (1, -1, 0) \).
Problem 10.31medium✓ Every equation proved
Find the tangent plane to \( \displaystyle z = x^{2} + y^{2} \) at the point \( \displaystyle (2, 1, 5) \).
Problem 10.32medium✓ Every equation proved
Find the tangent plane to \( \displaystyle z = x^{2} + y^{2} \) at the point \( \displaystyle (1, 1, 2) \).
Problem 10.33medium✓ Every equation proved
Find the tangent plane to \( \displaystyle z = x^{2} - y^{2} \) at the point \( \displaystyle (1, 1, 0) \).
Problem 10.34medium✓ Every equation proved
Find the tangent plane to \( \displaystyle z = x y \) at the point \( \displaystyle (0, 0, 0) \).
Problem 10.35medium✓ Every equation proved
Find the tangent plane to \( \displaystyle z = x^{2} - y^{2} \) at the point \( \displaystyle (0, 0, 0) \).
Problem 10.36medium✓ Every equation proved
Find the tangent plane to \( \displaystyle z = x y \) at the point \( \displaystyle (1, 2, 2) \).
Problem 10.38medium✓ Every equation proved
Find the tangent plane to \( \displaystyle z = x^{2} - y^{2} \) at the point \( \displaystyle (1, 2, -3) \).
Problem 10.84medium✓ Every equation proved
Find the tangent plane to \( \displaystyle z = x^{2} - y^{2} \) at the point \( \displaystyle (2, 1, 3) \).
Problem 10.85medium✓ Every equation proved
Find the tangent plane to \( \displaystyle z = x y \) at the point \( \displaystyle (1, -1, -1) \).
Problem 10.86medium✓ Every equation proved
Find the tangent plane to \( \displaystyle z = e^{x - y} \) at the point \( \displaystyle (1, 1, 1) \).
Problem 10.89medium✓ Nihil obstat
Find the tangent plane to \( \displaystyle z = x y \) at the point \( \displaystyle (2, 1, 2) \).
Problem 10.90medium✓ Every equation proved
Find the tangent plane to \( \displaystyle z = x^{2} y + 1 \) at the point \( \displaystyle (0, 0, 1) \).
Problem 10.92medium✓ Every equation proved
Find the tangent plane to \( \displaystyle z = x^{2} y + 1 \) at the point \( \displaystyle (2, 1, 5) \).
Problem 10.186medium✓ Every equation proved
Find the tangent plane to \( \displaystyle z = e^{x - y} \) at the point \( \displaystyle (1, -1, e^{2}) \).
Problem 10.37hard✓ Every equation proved