Math 8 Practice Exam Problems

Disclaimer: A few problems from the recent material

1.

Find the general solution to the differential equation $$\frac{dy}{dx} + \frac{y}{x\ln x} = x$$.

2.

Find the equation of the tangent plane to the level surface of $f(x,y,z) = ye^{-x^2}\sin z$ at $(0,1,\pi/3)$.

3.

Suppose that $z = f(x,y)$ is a smooth real-valued function of two variables, and that $$\frac{\partial f}{\partial x}(1,1) = 3$$ and $$\frac{\partial f}{\partial y}(1,1) = -1$$. If $x = s^2$ and $y = s^3$, we may then view $z$ as a function of the single variable $s$. The value of $$\frac{dz}{ds}$$ at $s = 1$ is

4.

Find an equation of the curve $y = f(x)$ that passes through the point $(1,1)$ and intersects all level curves of the function $g(x,y) = x^4 + y^2$ at right angles.

5.

A ball is placed at the point $(1,2,3)$ on the surface $z= y^2 - x^2$. Give the direction in the $xy$-plane corresponding to the direction in which the ball will start to roll. Describe the path in the $xy$-plane which the ball will follow. At the point $(1,2,3)$ what is the maximum rate at which the ball is descending.

6.

Let $f(x,y)= x^4 + y^4 + x^2 - y^2$. Find and classify all critical points of $f$. Use the method of Lagrange multipliers to find the largest and smallest values of $f$ on the circle $x^2 + y^2 = 4$.

7.

Consider a function $z = f(x,y)$ which is defined an has partial derviatives of all orders for all $x$ and $y$. Suppose the function $f(x,b)$ has a local maximum at $x=a$ and the function $f(a,y)$ has a local minimum at $y=b$. Can one infer that the point $(a,b)$ is a critical point, saddle point, local maximum, local minimum?