3.4 First-Order Partial Derivatives

First-Order Partial Derivatives

Introduction

First-Order Partial Derivatives

Suppose we take out an ＄18,000 car loan at interest rate $r$ and we agree to pay off the loan in $t$ years. The monthly payment, in dollars, is

If you take a loan with an interest rate of $3\%$ (so $r = 0.03$) and will pay it off in $t=4$ years, what will your monthly payment be?

Write a formula for the trace $f(x,2)$ as a function of $x$. On the axes provided, draw the graph of the trace with $y=2$ around the point where $x=1$. Then sketch the tangent line at the point with $x=1$ on your graph.

Compute the partial derivative $f_x(1,2)$ and relate its value to the sketch you just made.

Write a formula for the trace $f(1,y)$ as a function of $y$. On the axes provided, draw the graph of the trace with $x=1$. Then sketch the tangent line at the point with $y=2$ on your graph.

Compute the partial derivative $f_y(1,2)$ and relate its value to the sketch you just made.

If $f(x,y) = 3x^3 - 2x^2y^5$, find the partial derivatives $f_x$ and $f_y$.

If $f(x,y) = \displaystyle\frac{xy^2}{x+1}$, compute $\partial f/\partial x$ and $\partial f/\partial y$.

If $g(r,s) = rs\cos(r)$, find the partial derivatives $g_r$ and $g_s$.

Find all possible first-order partial derivatives of $q(x,t,z) = \displaystyle \frac{x2^tz^3}{1+x^2}.$