Q1. Find (a) an equation of the plane containing the points (3, −1, 1), (4, 0, 2), and (6, 3, 1), and (b) the area of the triangle with these vertices.

Background

Topic: Vectors and Planes in 3D Space

This question tests your understanding of how to find the equation of a plane given three points in space, and how to compute the area of a triangle defined by those points using vector methods.

Key Terms and Formulas

• Vector subtraction:

• Cross product: gives a vector perpendicular to both and

• Equation of a plane: , where is a normal vector

• Area of triangle:

Step-by-Step Guidance

1. Label the points as , , and .

2. Find vectors and by subtracting coordinates: , .

3. Compute the cross product to get a normal vector to the plane.

4. Use the normal vector and point to write the equation of the plane: .

5. To find the area of the triangle, calculate the magnitude of the cross product and use the formula for the area.

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Q2. Find the point where the line intersects the plane .

Background

Topic: Intersection of Lines and Planes in 3D

This question tests your ability to find the intersection point between a parametric line and a plane by substituting the line's coordinates into the plane's equation.

Key Terms and Formulas

• Parametric equations: , ,

• Plane equation:

Step-by-Step Guidance

1. Substitute , , and from the line's parametric equations into the plane equation.

2. Solve the resulting equation for .

3. Plug the value of back into the parametric equations to find the intersection point .

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Q3. Show that the planes and are neither parallel nor perpendicular. Find the angle between these planes.

Background

Topic: Angles Between Planes

This question tests your understanding of how to determine the relationship between two planes using their normal vectors, and how to compute the angle between them.

Key Terms and Formulas

• Normal vector of a plane: coefficients of , , in the plane equation

• Dot product:

• Angle formula:

Step-by-Step Guidance

1. Identify the normal vectors: and .

2. Check if the normals are scalar multiples (for parallelism) or if their dot product is zero (for perpendicularity).

3. Compute the dot product .

4. Find the magnitudes and .

5. Set up the formula for and prepare to solve for .

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Q4. Let be the curve with parametric equations , , . Find (a) the point where intersects the -plane, (b) the unit tangent vector to at this point, and (c) a vector equation for the tangent line to at this point.

Background

Topic: Parametric Curves and Tangent Vectors

This question tests your ability to analyze parametric curves, find intersection points with coordinate planes, compute tangent vectors, and write equations for tangent lines.

Key Terms and Formulas

• Intersection with -plane: set

• Tangent vector:

• Unit tangent vector:

• Equation of tangent line:

Step-by-Step Guidance

1. Set and solve for to find the intersection point.

2. Plug this value of into and to get the coordinates of the intersection point.

3. Compute the derivatives , , and to get the tangent vector at this .

4. Find the magnitude of the tangent vector and write the unit tangent vector.

5. Set up the vector equation for the tangent line at this point.

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Q5. Use traces to sketch the surface .

Background

Topic: Quadric Surfaces and Traces

This question tests your ability to analyze and sketch surfaces in three dimensions by examining their traces (cross-sections) in coordinate planes.

Key Terms and Formulas

• Trace: intersection of the surface with a plane (e.g., , , )

• Recognize the surface type: hyperboloid of one sheet

Step-by-Step Guidance

1. Set and describe the resulting curve in the -plane.

2. Set and describe the resulting curve in the -plane.

3. Set and describe the resulting curve in the -plane.

4. Use these traces to help sketch the overall surface.

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Q6. A particle moves in space with position , .

• (a) What is its initial position?

• (b) Find its velocity after time . What is its initial velocity?

• (c) Find its speed after time . What is its initial speed?

• (d) Find its acceleration after time . What is its initial acceleration?

Background

Topic: Vector Functions, Velocity, Acceleration, and Speed

This question tests your ability to differentiate vector-valued functions and interpret the results as velocity, speed, and acceleration.

Key Terms and Formulas

• Velocity:

• Speed:

• Acceleration:

Step-by-Step Guidance

1. Evaluate for the initial position.

2. Differentiate each component of to find , then evaluate at for initial velocity.

3. Compute the magnitude of for speed, and evaluate at for initial speed.

4. Differentiate to get , and evaluate at for initial acceleration.

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Q7. A moving particle starts at with initial velocity . Its acceleration at time is . Find its velocity and position at time .

Background

Topic: Motion with Variable Acceleration

This question tests your ability to integrate vector-valued acceleration functions to find velocity and position, using initial conditions.

Key Terms and Formulas

• Velocity:

• Position:

• Use initial conditions to solve for constants

Step-by-Step Guidance

1. Integrate each component of to find , introducing constants of integration.

2. Use the initial velocity to solve for these constants.

3. Integrate to find , again introducing constants.

4. Use the initial position to solve for these constants.

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Q8. Find the length of the parametrized curve on the interval .

Background

Topic: Arc Length of Space Curves

This question tests your ability to compute the arc length of a curve given by a vector function over a specified interval.

Key Terms and Formulas

• Arc length:

• Given:

Step-by-Step Guidance

1. Compute by differentiating each component.

2. Find and simplify the expression.

3. Set up the definite integral for arc length from to .

4. Use the provided integral formula to help evaluate the integral.

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Q9. Let .

• (a) Find and sketch the domain of .

• (b) Find all points where the tangent plane to the graph of is horizontal.

Background

Topic: Multivariable Functions – Domains and Tangent Planes

This question tests your understanding of domains of functions of two variables and the conditions for a tangent plane to be horizontal.

Key Terms and Formulas

• Domain: set where the argument of is positive

• Tangent plane is horizontal when and

Step-by-Step Guidance

1. Set to find the domain.

2. Find and and set them equal to zero to find points where the tangent plane is horizontal.

3. Solve the resulting equations for .

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Q10. Show that the following limit does not exist: .

Background

Topic: Limits in Several Variables

This question tests your ability to analyze limits in two variables and show non-existence by approaching along different paths.

Key Terms and Formulas

• Limit along a path: substitute or

• If the limit depends on the path, it does not exist

Step-by-Step Guidance

1. Choose two different paths approaching (e.g., and ).

2. Substitute each path into the function and simplify.

3. Compute the limit along each path and compare the results.

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Q11. Let .

• (a) Calculate the directional derivative of at the point in the direction of .

• (b) Give a unit vector in the plane that points in the direction of steepest ascent on the graph at the point . What is the directional derivative of at this point in this direction?

• (c) Give two different unit vectors that point in directions of no change along the graph of at the point .

• (d) Find an equation of the tangent plane to the surface at .

• (e) Use your answer in part (d) to estimate . Give your answer as a fraction of integers.

• (f) Verify that .

Background

Topic: Partial Derivatives, Directional Derivatives, and Tangent Planes

This question tests your ability to compute and interpret partial derivatives, directional derivatives, and tangent planes for functions of two variables.

Key Terms and Formulas

• Gradient:

• Directional derivative: (with a unit vector)

• Tangent plane:

• Mixed partials: and

Step-by-Step Guidance

1. Compute the gradient and evaluate at the required points.

2. Normalize direction vectors as needed.

3. Use the formulas for directional derivatives and tangent planes as appropriate.

4. For mixed partials, compute and and compare.

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Q12. Use differentials to estimate the change in as changes from to . Give your answer as a fraction of integers.

Background

Topic: Differentials and Linear Approximation

This question tests your ability to use differentials to estimate small changes in a function of two variables.

Key Terms and Formulas

• Differential:

• For , ,

Step-by-Step Guidance

1. Compute and as the changes in and .

2. Find and at the point .

3. Plug these values into the differential formula to estimate .

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Q13. Find the directional derivative of at in the direction of .

Background

Topic: Directional Derivatives

This question tests your ability to compute the directional derivative of a function at a point in a specified direction.

Key Terms and Formulas

• Gradient:

• Directional derivative: , with a unit vector in the direction of

Step-by-Step Guidance

1. Compute the gradient at .

2. Normalize to get a unit vector .

3. Compute the dot product to get the directional derivative.

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Q14. Let .

• (a) Find the domain and range of .

• (b) Sketch the traces of the surface on the , , and planes.

• (c) Find the equation of the tangent plane to this surface at .

• (d) Sketch the surface and its tangent plane at this point.

Background

Topic: Quadratic Surfaces and Tangent Planes

This question tests your understanding of domains, ranges, traces, and tangent planes for surfaces defined by functions of two variables.

Key Terms and Formulas

• Domain: all for which is defined

• Range: all possible values

• Tangent plane:

Step-by-Step Guidance

1. Analyze the formula for to determine domain and range.

2. Set , , or to find traces in the coordinate planes.

3. Compute and at and use them to write the tangent plane equation.

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Q15. Calculate the tangent planes to the surface defined by at the points and .

Background

Topic: Implicit Differentiation and Tangent Planes

This question tests your ability to find tangent planes to implicitly defined surfaces using partial derivatives.

Key Terms and Formulas

• Implicit surface:

• Tangent plane:

Step-by-Step Guidance

1. Define .

2. Compute the partial derivatives , , and .

3. Evaluate these derivatives at each given point.

4. Write the equation of the tangent plane at each point using the formula above.

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