MATH 2551-C/HP– Multivariable Calculus
Spring 2025
Full name:
GT ID:
Quiz 5 Version C
You have 10 minutes to take the quiz. You may not use aids of any kind. The Learning Outcomes
associated with this Quiz are printed on the back page along with some relevant formulas.
1. [D2: Interpreting Derivatives] Choose whether the following statement is true or false. If the
statement is always true, pick true. If the statement is ever false, pick false.
If f (x, y) is differentiable at (a, b) with f (a, b) = c, then −∇f (a, b) is orthogonal to the curve
f (x, y) = c.
√
TRUE
⃝ FALSE
Solution. This is true - the gradient vector at a point is always orthogonal to the level set of
f containing that point and therefore so is its negative.
2. [D3: Tangent Planes and Linear Approximations] Find an equation for the tangent plane
to the graph of the surface
x2 − y 4 + ez − 2z = 1
at the point (1, 1, 0).
Solution. This surface is a level surface for the function of three variables
f (x, y, z) = x2 − y 4 + ez − 2z,
so a normal vector to the surface and tangent plane at this point is ∇f (1, 1, 0). We have
∇f = ⟨2x, −4y 3 , ez − 2⟩,
so ∇f (1, 1, 0) = ⟨2, −4, −1⟩ and therefore the tangent plane is
2(x − 1) − 4(y − 1) − (z − 0) = 0
or
2x − 4y − z = −2.
MATH 2551-C/HP– Multivariable Calculus
Spring 2025
Quiz 5 Learning Outcomes
• D2: Interpreting Derivatives. Interpret the meaning of the gradient and directional deriva­
tive of a function at a given point in a specified direction, including in the context of a graph or
a contour plot.
• D3: Tangent Planes and Linear Approximations. Find equations for tangent planes to
surfaces and linear approximations of functions at a given point and use these to solve problems.
Useful Formulas
• Linearization: Near a, f (x) ≈ L(x) = f (a) + Df (a)(x − a)
• The tangent line to a level curve of f (x, y) at (a, b) is
0 = ∇f (a, b) · ⟨x − a, y − b⟩.
• The tangent plane to a level surface of f (x, y, z) at (a, b, c) is
0 = ∇f (a, b, c) · ⟨x − a, y − b, z − c⟩.