Vectors
A force F = 2i + 6j moves an object from P(2, 2) to Q(4, 6). How much work is done?
28 foot pounds
Simplify sin(arccos(x)), for −3 < x < 3, to an algebraic expression.
√(1 − x²)
Evaluate lim (t → 1) (1 − t²) / (1 − √t)
Answer: 4
The displacement (in feet) of a particle moving in a straight line is given by s(t) = t² + 5t − 6, where t is measured in seconds. Find the average velocity over the interval [1, 4]
10 ft/s
Given a = (2, −1) and b = (1, −3), find the scalar projection of b onto a.
5/sqrt(5)
For points A(1, 3), B(−3, 1), and C(2, 1), which of the following statements is false?
(a) AB = 〈−4, −2〉
(b) cos θ = (〈5, 0〉 · 〈−4, −2〉) / (5√20), where θ = ∠ABC
(c) |AC| = √5
(d) |AB| = √20
(e) AB is perpendicular to AC
The following is false:
(b) cos θ = (〈5, 0〉 · 〈−4, −2〉) / (5√20), where θ = ∠ABC
Simplify cos(arcsin(x/3)).
√(9 − x²)/3
Find the limit lim (x → −4⁻) [ x / (x + 4) ]
Answer: ∞
Given f(x) = 1/x and f′(x) = −1/x², find the equation of the tangent line of f(x) at x = 2.
y − 1/2 = −1/4 (x − 2)
Find the point of intersection of the following two lines, if it exists:
L₁(t) = (7 − 3t, 4+t)
L₂(s) = (1 + s, 2 + s)
(4, 5)
Find the vector a that has magnitude |a| = 6 and makes an angle of 300° with the positive x-axis.
(a) 3i + 3√3 j
(b) 3√3 i + 3j
(c) −3√3 i − 3j
(d) 3√3 i − 3j
(e) 3i − 3√3 j
Answer: (e) 3i − 3√3 j
Evaluate lim (x → −∞) arctan(5/x²).
Find the average rate of change of f(t) = √(2t + 3) from t = 1 to t = 3.
Answer: (3 − √5)/2
Given f(x) = x³ − 3x + 1 and f′(x) = 3x² − 3, find the equation of the tangent line to f(x) at x = −2.
y = 9x + 17
Find the distance between the point (−2, 5) and the line y = x + 1.
3/√2
Find a vector equation for the line that passes through the point (2, −1) and is perpendicular to 〈3, 4〉.
(a) r(t) = 〈2 + 3t, −1 + 4t〉
(b) r(t) = 〈2 + 4t, −1 + 3t〉
(c) r(t) = 〈2 − 4t, −1 + 3t〉
(d) r(t) = 〈2 − 3t, −1 − 4t〉
(e) r(t) = 〈1 − 4t, −2 + 3t〉
Answer: (c) r(t) = 〈2 − 4t, −1 + 3t〉
Which of the following is equal to sin(arctan(2x))?
(a) 2x / √(1 + x²)
(b) 2x / (1 + x²)
(c) 2x / √(1 + 2x²)
(d) 1 / √(1 + 4x²)
(e) 2x / √(1 + 4x²)
(e) 2x / √(1 + 4x²)
Which of the following intervals contains a root to the equation x³ + 2x² = 42?
(a) (2, 3)
(b) (1, 2)
(c) (0, 1)
(d) (−1, 0)
(e) (−2, −1)
(a) (2, 3)
Use the definition of the derivative to find f′(x) for f(x) = √(8x − x²).
(No shortcut rules are allowed.)
f′(x) = (4 − x) / √(8x − x²)
The motion of a particle is given by the vector function
r(t) = ⟨2 cos t, −3 sin t⟩.
Which of the following describes the motion of the particle as t increases?
(a) Clockwise around a circle
(b) Counterclockwise around an ellipse
(c) Counterclockwise around a circle
(d) Clockwise around an ellipse
(e) None of these
(d) Clockwise around an ellipse
Find the distance from the point (1, 5) to the line y = 2x + 1.
Answer: (a) 2/√5
Find the cosine of the angle between the vectors (−4, 2) and (1, 5).
cos θ = −6 / (√20 · √26)
Find the horizontal and vertical asymptotes for f(x) = (2 − x)(3x + 1) / (x² − 4)
Answer: y = −3, x = −2
Use the definition of the derivative to find f′(x) for f(x) = 3 / (x − 2).
(No points will be given for shortcut formulas.)
f′(x) = −3 / (x − 2)²
A pilot steers a plane in the direction 210° counterclockwise from the positive x-axis at 400 mph.
The wind blows at 20 mph in the direction 60° counterclockwise from the positive x-axis.
Find the resultant velocity vector of the plane.
(−200√3 − 10, −200 + 10√3) mph