Home›Calculus 3›Tangent planes

Tangent planes practice problems

The plane that touches a surface z = f(x, y) at a point. 24 problems with worked solutions; in 24 of them every equation is proved by a computer algebra system.

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 \( \displaystyle (0, 0, 1) \).

Problem 10.87medium✓ Every equation proved

Find the tangent plane to \( \displaystyle z = e^{x - y} \) at the point \( \displaystyle (1, 1, 1) \).

Problem 10.89medium✓ Every equation proved

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 \( \displaystyle z = e^{x - y} \) at the point \( \displaystyle (2, 1, e) \).

Problem 10.91medium✓ 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 \( \displaystyle z = e^{x - y} \) at the point \( \displaystyle (1, -1, e^{2}) \).

Problem 10.37hard✓ Every equation proved

Find the tangent plane to \( \displaystyle z = e^{x - y} \) at the point \( \displaystyle (1, 2, e^{-1}) \).

Problem 10.88hard✓ Every equation proved