Mathematics-I Exam Prep
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Mathematics-I Exam Prep
Sample questions with model answers
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Find the equation of the tangent plane and normal line to the surface $\frac{x^2}{4} + \frac{y^2}{9} + \frac{z^2}{16} = 3$ at $(-2, 1, -3)$.
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Use the gradient to find the tangent plane and normal line.
The full exam-length answer is in the app.
Find the extreme values of $f(x,y) = x^3 + y^3 - 3x - 12y + 20$.
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Use critical points and the second derivative test.
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Find the equation of the tangent plane and normal line to the surface $xy^2 + 2yz^2 + 3zx^2 = 6$ at the point $(1, 1, 1)$.
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Calculate the gradient at the point to determine the plane and line.
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Find the equation of the tangent plane and normal line to the surface $x^2 + y^2 + z - 9 = 0$ at the point $(1, 2, 4)$.
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Determine the gradient and use it to find the plane and line.
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Find the extreme values of $f(x,y) = x^3 + y^3 - 3axy$ ($a > 0$).
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Analyze critical points for local extrema.
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Define gradient and directional derivative, and find the directional derivative of $f(x,y,z) = x^2yz + 4xz^2$ at $(1, -2, -1)$ in the direction of $2\hat{i} - \hat{j} - 2\hat{k}$.
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Compute the gradient and project it onto the direction vector.
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Find the directional derivative of $f(x,y,z) = x^2 - 2y^2 + 4z^2$ at the point $(1, 1, -1)$ in the direction of the vector $2\hat{i} + \hat{j} - \hat{k}$.
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Evaluate the directional derivative using the gradient.
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Find the directional derivative of $\phi = x^2yz + 4xz^2$ at $(1, -2, 1)$ in the direction of the normal to the surface $f = x \log z - y^2$ at $(-1, 2, 1)$.
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Calculate the normal vector and evaluate the directional derivative.
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Find the values of the constants $a, b, c$ so that the directional derivative of $\phi = axy^2 + byz + cz^2x^2$ at $(1, 2, -1)$ has maximum magnitude 64 in a direction parallel to the z-axis.
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Set up the conditions for maximum directional derivative.
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In what direction from the point $(1, 3, 2)$ is the directional derivative of $\phi = 2xz - y^2$ a maximum? Find also the magnitude of this maximum.
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Use the gradient to find the direction and magnitude.
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Define local extreme values for a function of two variables and find the local extreme values of $x^3 + 3xy^2 - 3x^2 - 3y^2 + 4$.
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Identify critical points and use the second derivative test.
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Examine the minimum and maximum value of $f(x,y) = \sin x + \sin y + \sin(x+y)$.
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Analyze the function over its domain.
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A delivery company accepts only rectangular boxes for which the sum of the length and girth (perimeter of a cross section) does not exceed 108 inches. Find the dimensions of an acceptable box of largest volume, using Lagrange's multiplier method.
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Apply Lagrange's multipliers to maximize volume.
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Find a point on the plane $2x + 3y - z = 5$ which is nearest to the origin.
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Minimize the distance function subject to the plane constraint.
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Find the shortest and largest distance from the point $(1, 2, -1)$ to the sphere $x^2 + y^2 + z^2 = 24$.
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Calculate distances from the point to the sphere.
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