Suppose the quantities $x$, $y$, $z$ and $t$ satisfy $z = f(x, y)$, $x = g(t)$, and $y = h(t)$. Given g(7) = 2, $g'(7) = 3$, h(7) = -1, $h'(7) = -2$, $f_x(2, -1) = -2$ and $f_y(2, -1) = 2$, compute $\frac{dz}{dt}$ at $t = 7$.
Added by Brandon C.
Close
Step 1
Since z = f(x, y), x = g(t), and y = h(t), we can write z as a function of t: z(t) = f(g(t), h(t)). Using the chain rule, we have: dz/dt = ∂z/∂x * dx/dt + ∂z/∂y * dy/dt Show more…
Show all steps
Your feedback will help us improve your experience
Sri K and 66 other Calculus 1 / AB educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
Find parametric equations for the tangent line to the curve of intersection of the paraboloid z = x^2 + y^2 and the ellipsoid 5x^2 + 4y^2 + 2z^2 = 17 at the point (-1, 1, 2). (Enter your answer as a comma-separated list of equations. Let x, y, and z be in terms of t.)
Sri K.
Calculate dz/dt given the following functions. Express the final answer in terms of t. z = f(x,y) = x^2 + xy + y^2, x = x(t) = 7t, y = y(t) = e^t
Urvashi A.
Consider the system of differential equations: y1' = 5y1 + 3y2, y2' = 3y1 + 5y2. Verify that for any constants c1 and c2, the functions: y1(t) = c1e^2t + c2e^8t, y2(t) = -c1e^2t + c2e^8t satisfy the system of differential equations. Enter c1 as c1 and c2 as c2. a. Find the value of each term in the equation y1' = 5y1 + 3y2 in terms of the variable t. (Enter the terms in the order given.) b. Find the value of each term in the equation y2' = 3y1 + 5y2 in terms of the variable t. (Enter the terms in the order given.)
Sam S.
Recommended Textbooks
Calculus: Early Transcendentals
Thomas Calculus
Transcript
18,000,000+
Students on Numerade
Trusted by students at 8,000+ universities
Watch the video solution with this free unlock.
EMAIL
PASSWORD