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Multivariable Calculus · Multivariable Differentiation

Tangent Planes — Practice Problems and Worked Solutions

Build the best local linear surface.

The formulas you need here

Tangent plane

z = f (a, b) + f_{x} (a, b) (x - a) + f_{y} (a, b) (y - b)

What it says: The tangent line idea, with one term for each input direction.
When to reach for it: Linear approximation of a surface near a point.
Watch out: Both partials are evaluated at the point before being used as coefficients.

z = 1 + 2 (x - 1) + 2 (y - 1)

Tangent to z = x² + y² at (1,1).

17 worked problems

Every step is generated and checked by a computer algebra system.

1.
Find the tangent plane, written as z = ...
$z = x^{2} + y^{2}, (x, y) = (1, 1)$
Answer: $2 x + 2 y - 2$
Step-by-step solution
1. $z = f (a, b) + f_{x} (a, b) (x - a) + f_{y} (a, b) (y - b)$
2. $z = 2 x + 2 y - 2$
2.
Find the tangent plane, written as z = ...
$z = x y, (x, y) = (2, 1)$
Answer: $x + 2 y - 2$
Step-by-step solution
1. $z = f (a, b) + f_{x} (a, b) (x - a) + f_{y} (a, b) (y - b)$
2. $z = x + 2 y - 2$
3.
Find the tangent plane, written as z = ...
$z = x^{2} - y, (x, y) = (- 1, 2)$
Answer: $- 2 x - y - 1$
Step-by-step solution
1. $z = f (a, b) + f_{x} (a, b) (x - a) + f_{y} (a, b) (y - b)$
2. $z = - 2 x - y - 1$
4.
Find the tangent plane, written as z = ...
$z = y + e^{x}, (x, y) = (0, 1)$
Answer: $x + y + 1$
Step-by-step solution
1. $z = f (a, b) + f_{x} (a, b) (x - a) + f_{y} (a, b) (y - b)$
2. $z = x + y + 1$
5.
Find the tangent plane, written as z = ...
$z = x^{3} + x y, (x, y) = (1, - 1)$
Answer: $2 x + y - 1$
Step-by-step solution
1. $z = f (a, b) + f_{x} (a, b) (x - a) + f_{y} (a, b) (y - b)$
2. $z = 2 x + y - 1$
6.
Find the tangent plane, written as z = ...
$z = y^{2} + sin (x), (x, y) = (0, 2)$
Answer: $x + 4 y - 4$
Step-by-step solution
1. $z = f (a, b) + f_{x} (a, b) (x - a) + f_{y} (a, b) (y - b)$
2. $z = x + 4 y - 4$
7.
Find the tangent plane, written as z = ...
$z = x^{2} + 2 y^{2}, (x, y) = (1, 1)$
Answer: $2 x + 4 y - 3$
Step-by-step solution
1. $z = f (a, b) + f_{x} (a, b) (x - a) + f_{y} (a, b) (y - b)$
2. $z = 2 x + 4 y - 3$
8.
Find the tangent plane, written as z = ...
$z = x y + x, (x, y) = (2, 1)$
Answer: $2 x + 2 y - 2$
Step-by-step solution
1. $z = f (a, b) + f_{x} (a, b) (x - a) + f_{y} (a, b) (y - b)$
2. $z = 2 x + 2 y - 2$
9.
Find the tangent plane, written as z = ...
$z = x^{2} - 2 y, (x, y) = (1, 3)$
Answer: $2 x - 2 y - 1$
Step-by-step solution
1. $z = f (a, b) + f_{x} (a, b) (x - a) + f_{y} (a, b) (y - b)$
2. $z = 2 x - 2 y - 1$
10.
Find the tangent plane, written as z = ...
$z = x^{3} + y^{2}, (x, y) = (1, 2)$
Answer: $3 x + 4 y - 6$
Step-by-step solution
1. $z = f (a, b) + f_{x} (a, b) (x - a) + f_{y} (a, b) (y - b)$
2. $z = 3 x + 4 y - 6$
11.
Find the tangent plane, written as z = ...
$z = x y^{2}, (x, y) = (2, 1)$
Answer: $x + 4 y - 4$
Step-by-step solution
1. $z = f (a, b) + f_{x} (a, b) (x - a) + f_{y} (a, b) (y - b)$
2. $z = x + 4 y - 4$
12.
Find the tangent plane, written as z = ...
$z = y^{2} + e^{x}, (x, y) = (0, 2)$
Answer: $x + 4 y - 3$
Step-by-step solution
1. $z = f (a, b) + f_{x} (a, b) (x - a) + f_{y} (a, b) (y - b)$
2. $z = x + 4 y - 3$
13.
Find the tangent plane, written as z = ...
$z = x^{2} y, (x, y) = (1, 1)$
Answer: $2 x + y - 2$
Step-by-step solution
1. $z = f (a, b) + f_{x} (a, b) (x - a) + f_{y} (a, b) (y - b)$
2. $z = 2 x + y - 2$
14.
Find the tangent plane, written as z = ...
$z = 2 x + y^{3}, (x, y) = (3, 1)$
Answer: $2 x + 3 y - 2$
Step-by-step solution
1. $z = f (a, b) + f_{x} (a, b) (x - a) + f_{y} (a, b) (y - b)$
2. $z = 2 x + 3 y - 2$
15.
Find the tangent plane, written as z = ...
$z = x^{2} + y, (x, y) = (2, 4)$
Answer: $4 x + y - 4$
Step-by-step solution
1. $z = f (a, b) + f_{x} (a, b) (x - a) + f_{y} (a, b) (y - b)$
2. $z = 4 x + y - 4$
16.
Find the tangent plane, written as z = ...
$z = y + cos (x), (x, y) = (0, 3)$
Answer: $y + 1$
Step-by-step solution
1. $z = f (a, b) + f_{x} (a, b) (x - a) + f_{y} (a, b) (y - b)$
2. $z = y + 1$
17.
Find the tangent plane, written as z = ...
$z = x y + y^{2}, (x, y) = (1, 2)$
Answer: $2 x + 5 y - 6$
Step-by-step solution
1. $z = f (a, b) + f_{x} (a, b) (x - a) + f_{y} (a, b) (y - b)$
2. $z = 2 x + 5 y - 6$

The formulas you need here

Tangent plane

17 worked problems

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