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MATH2500 Exam 1
| Question | Answer |
|---|---|
| Find the unit tangent vector T(t) at the point with the given value of the parameter t | 1. Take derivative of given r(t) 2. Plug in parameter into found derivative 3. Multiply that by 1/magnitude of derivative (length) |
| Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point | |
| Find the length of the given curve | Use arc length equation. Integral of the square root of the derivatives squared |
| Find and sketch the level curves f(x, y) = c on the same set of coordinate axes for the given values of c | |
| Find an equation of the tangent plane to the given surface at the specified point | Use tangent plane to the surface equation. F(a,b)+Fx(a,b)(x-a)+Fy(a,b)(y-b) |
| Find the linearization L(x, y) of f at a point | Use tangent plane to the surface equation except change z= to L= and plug in values trying to approximate |
| Find the directional derivative of the function at the given point in the direction of the vector v. | 1. Normalize v to obtain a unit vector u in the direction of v 2. Compute gradient for the equation at the point |
| Define y as a differentiable function of x . Find the values of dx/dy at the given point | 1. Differentiate implicitly wrt x 2. Solve for dy/dx 3. Plug in the given point |
| Find the maximum rate of change of f at the given point and the direction in which it occurs | 1. Find the gradient at the given point 2. Find max rate of change by finding the magnitude of the gradient 3. To find direction, solve for the gradient of f over magnitude of gradient of f |
| What is the discriminant | fxxfyy-(fxy)^(2) |
| When does a local maximum occur | When D> 0 and fxx<0 |
| When does a local minimum occur | When D>0 and fxx>0 |
| When does a saddle point occur | When D<0 |
| When is the second derivative test inconclusive | When D=0 |
| Use Lagrange multipliers to find the maximum and minimum values of f subject to the given constraints | 1. Find partial derivatives for equation and constraint, 2. Use formula gradientf=gradientg* lambda, 3. Set equations to solve for lambda, 4. Set lambda equations equal to isolate y or x, 5. Plug back into constraint equation to find other y or x, |
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ephemeral