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Placement Test Practice — Multivariable Calculus
This practice test covers partial derivatives, multiple integrals, gradient and directional derivatives, and line integrals. Apply the chain rule for multivariable functions, set up iterated integrals with correct bounds, compute gradients, and evaluate line integrals using parameterization or the fundamental theorem for conservative fields.
Placement Test Practice — Multivariable Calculus
Practice Test — 20 Questions
1. Find $f_x$: $f = x^3y^2$.
2. Find $f_y$: $f = \sin(2x+y)$.
3. Evaluate $\int_0^1\int_0^2 3xy\,dy\,dx$.
4. Find $\nabla f$ at $(1,1)$: $f = x^2+xy$.
5. Critical point of $f = x^2+y^2-6x$?
6. $D_{\mathbf{u}}f$ at $(2,0)$: $f = x^2-y^2$, $\mathbf{u} = \langle 0,1 \rangle$.
7. Is $\langle y^2, 2xy \rangle$ conservative?
8. Evaluate $\int_0^2\int_0^x 1\,dy\,dx$.
9. Max rate of change at $(3,4)$: $f = \sqrt{x^2+y^2}$?
10. $f_{xy}$ for $f = x^3y^2$?
11. Classify $(0,0)$: $f = xy$.
12. Potential of $\langle 2x, 2y \rangle$?
13. $\int_0^1\int_0^1\int_0^1 1\,dz\,dy\,dx$?
14. $\nabla f$ for $f = e^{x+y}$ at $(0,0)$?
15. What does the gradient point toward?
16. $f_x + f_y$ for $f = x+y$?
17. $\oint_C \nabla f \cdot d\mathbf{r}$ for any closed $C$?
18. Evaluate $\int_0^{\pi/2}\int_0^1 r\,dr\,d\theta$.
19. Saddle point means $D$ is ___.
20. $\int_C \mathbf{F}\cdot d\mathbf{r}$ from $(0,0)$ to $(1,1)$: $\mathbf{F} = \nabla(xy)$.
Show Answer Key
1. $3x^2y^2$
2. $\cos(2x+y)$
3. $\int_0^1 6x\,dx = 3$
4. $\langle 3, 1 \rangle$
5. $(3,0)$
6. $\nabla f(2,0) = \langle 4,0 \rangle$; $D_{\mathbf{u}} = 0$
7. $P_y = 2y = Q_x$; yes
8. $\int_0^2 x\,dx = 2$
9. $|\nabla f| = 1$
10. $6x^2y$
11. $D = 0 \cdot 0 - 1 = -1 < 0$; saddle
12. $f = x^2+y^2+C$
13. $1$
14. $\langle 1,1 \rangle$
15. Direction of steepest ascent
16. $2$
17. $0$
18. $\frac{\pi}{4}$
19. Negative ($D < 0$)
20. $f(1,1)-f(0,0) = 1$