5 / 5
Placement Test Practice — Multivariable Calculus
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. $f(1,1)-f(0,0) = 1$