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You CAN Ace Calculus

17calculus > exam list > calc3 exam A1

Calculus Main Topics

Single Variable Calculus

Multi-Variable Calculus

Tools

math tools

general learning tools

other exams from this semester

exam A1 - exam A2 - exam A3 - exam A4

complete exam list

free ideas to save on books - bags - supplies ATTENTION INSTRUCTORS: The new 2018 version of 17calculus will include changes to the practice problem numbering system. If you would like advance information to help you prepare for spring semester, send us an email at 2018info at 17calculus.com.

Calculus 3 - Exam 1 ( Semester A )

This is the first exam for third semester calculus.

Exam Details

Tools

Time

2 hours

Calculators

no

Questions

5

Formula Sheet(s)

1 page, 8.5x11 or A4

Total Points

75

Other Tools

none

Instructions:
- For each problem, correct answers are worth 1 point. The remaining points are earned by showing calculations and giving reasoning that justify your conclusions.
- Correct notation counts (i.e. points will be taken off for incorrect notation).
[ Click on the question to reveal/hide the solution. ]

Question 1

(20 points) For $$\vec{r}(t) = 2\cos(t^2) \hat{i} + 2\sin(t^2)\hat{j}$$, $$t\ge0$$, compute
a. $$\vec{T}(t)$$; b. $$\vec{N}(t)$$; c. curvature; d. $$a_T$$ and $$a_N$$; e. $$\vec{r}(s)$$

solution

Question 2

(5 points) Compute the indefinite integral of $$\vec{r}(t)= (5t^{-4}-t^2)\hat{i} + (t^6-4t^3)\hat{j} + (2/t)\hat{k}$$

solution

Question 3

(10 points) For the trajectory $$\vec{r}(t)=\langle e^t\sin(t), e^t\cos(t), e^t\rangle$$, find
a. the speed associated with the trajectory;
b. the length of the trajectory on the interval $$0\leq t \leq \ln(2)$$

solution

Question 4

(15 points) For the vectors $$\vec{u}=\langle 1,3,-2 \rangle$$ and $$\vec{v}=\langle 4,2,1 \rangle$$, find
a. the cosine of the angle $$\theta$$ between the vectors
b. the projection of $$\vec{v}$$ onto $$\vec{u}$$; c. $$\vec{u} \times \vec{v}$$

solution

Question 5

(25 points) A baseball is hit 3 ft above home plate with an initial velocity of $$\langle 65,85,85\rangle$$ ft/sec. The spin on the baseball produces a horizontal acceleration of the ball of 12 ft/sec2 in the eastward direction. Assume the x-axis points east, the y-axis points north and the z-axis is vertical with g=32 ft/sec2.     a. Find the velocity vector for $$t\ge0$$.     b. Find the position vector for $$t\ge0$$.
c. Clearly explain how you would determine the time of flight and range of the ball (do not calculate).
d. Clearly explain how you would determine the maximum height of the ball (do not calculate).

solution