Theorems
How do you correctly justify the Mean Value Theorem?
What is, "since F is stated to be differentiable, and therefore continuous, there must be a value c on the interval [a,b] such that F'(c) = [F(b) - F(a)] / b-a."
Which of the following statements is true for the function F(x) = (lnx)/x, x > 0?
A. F is increasing on the interval (0,∞)
B. F is increasing on the interval [1,∞)
C. F is decreasing on the interval [1,e]
D. F is decreasing on the interval [e,∞)
What is D. F is decreasing on the interval [e,∞)
F'(x) = (1-lnx)/x2
F'(e) = 0
F'(x) < 0 for x > e
For what values x does the graph of y = 3x5 + 10x4 have a point of inflection, and what is it's concavity before and after said point?
A.) x= -8/3 only
B.) x = -2 only
C.) x = 0 only
D.) x = 0 only and x= -8/3
E.) x = 0 only and x = -2
What is B. x = -2 and concave down to concave up. y'' = 0 at x = -2 and 0, but y'' doesn't change signs at x = 0. y''< 0 when x < -2, and y'' > 0 when x > -2
In the xy-plane, how many points on the curve y2+x2=3−xy have horizontal or vertical tangent lines?
A. No points have vertical tangent lines, and two points have horizontal tangent lines.
B. One point has a vertical tangent line, and one point has a horizontal tangent line.
C. Two points have vertical tangent lines, and two points have horizontal tangent lines.
D. No points have vertical tangent lines, and no points have horizontal tangent lines.
What is C.
2yy′+2x=−xy′−y -> y′=(−2x−y)/(2y+x)
The AP Exam...
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Let G be a differentiable function where G(-5) =10 and G(-3) = 2. Is there a number c in the closed interval [-5, -3] such that g'(c) = -4? Justify your answer.
What is yes. Since G is stated to be differentiable, and therefore continuous, there must be a value c on the interval [-5,-3] such that G'(c) = [G(-3) - G(-5)]/-3- (-5). (2-10)/(-3-(-5)) = -4
The acceleration, in centimeters per second per second, of a projectile is modeled by A(t)=2t3−15t2+36t, where t is measured in seconds. What is the projectile’s maximum acceleration on the time interval 0≤t≤4 ?
A. The maximum acceleration of the projectile is 4 centimeters per second per second and occurs at t=32 seconds.
B. The maximum acceleration of the projectile is 27 centimeters per second per second and occurs at t=3 seconds.
C. The maximum acceleration of the projectile is 28 centimeters per second per second and occurs at t=2 seconds.
D. The maximum acceleration of the projectile is 32 centimeters per second per second and occurs at t=4 seconds.
What is D.
A′(t) = 6t2−30t+36 = 6(t2−5t+6) = 6(t−2)(t−3)
A'(t) = 0 at t=2 and t=3.
A(0)=0
A(2)=28
A(3)=27
A(4)=32
Let F be the function defined by F(x)=xcosx − sinx. What is the absolute maximum value of F on the interval [−π/2,2π]?
A. -π
B. 2π
C. 0
D. 1
What is B
F′(x) = −xsinx = 0 at x=0, x=π, and x=2π
f(−π/2)=1
f(0)=0
f(π)=−π
f(2π)=2π
In the xy-plane, the point (0,2) is on the curve C. If dy/dx=−4x/3y for the curve, which of the following statements is true?
A. At the point (0,2), the curve C has a relative minimum because dy/dx=0 and d2y/dx2>0.
B. At the point (0,2), the curve C has a relative minimum because dy/dx=0 and d2y/dx2<0.
C. At the point (0,2), the curve C has a relative maximum because dy/dx=0 and d2y/dx2>0.
D. At the point (0,2), the curve C has a relative maximum because dy/dx=0 and d2y/dx2<0.
What is D.
d2y/dx2 = ((3y)(−4)−(−4x)(3dy/dx))/9y2
d2y/dx2 = -48/72
True or False
If F'(x) = 0 at x = c, then c is the location of a relative/absolute extrema on the graph of F(x).
If F''(x) = 0 at x = c, then c is the location of a point of inflection on the graph of F(x).
What is False
F'(x) must change signs at x =c for extrema to occur on the graph of F(x)
F''(x) must change signs at x =c for a point of inflection to occur on the graph of F(x)
Let F be the function given by F(x) = (x2 - 9)/(sinx) on the closed interval [0,5]. Of the following intervals, on which can the Mean Value Theorem be applied to F?
I. [1,3], because F is continuous on [1,3] and differentiable on (1,3).
II. [4,5], because F is continuous on [4,5] and differentiable on (4,5).
III. [1,4], because F is continuous on [1,4] and differentiable on (1,4).
A. None
B. I only
C. I and II only
D. I, II, and III
What is C. I and II only
The number of fish in a small bay is modeled by the function F defined by F(t)=10(t3−12t2+45t+100), where t is measured in days and 0 ≤ t ≤ 8.
What is the absolute minimum number of fish in the bay over the time interval, and for what values of t, 0 ≤ t ≤ 8, is the rate of change of the number of fish in the bay decreasing?
What is Fmin = 1,000 fish and F' is decreasing from 0 < t < 4. F'' < 0
The total cost, in dollars, to order x units of a certain product is modeled by C(x) = 7x2+252. According to the model, for what size order is the cost per unit a minimum?
A. An order of 1 unit has a minimum cost per unit.
B. An order of 6 units has a minimum cost per unit.
C. An order of 84 units has a minimum cost per unit.
D. An order of 252 units has a minimum cost per unit.
What is B
Cost = 7x2+252
Cost/unit = (7x2+252)/x = 7x +252/x
d/dx(7x +252/x) = 7 -252/x2 = 0 at x = 6
The point (1,1) is on the curve defined by x2+y3=2. Which of the following statements is true about the curve at the point (1,1)?
A. y' > 0 and y'' > 0
B. y' > 0 and y'' < 0
C. y' < 0 and y'' > 0
D. y' < 0 and y'' < 0
What is D. y' < 0 and y'' < 0.
y' = -2/3 and y'' = -14/9
The correct application of the chain rule for dy/dx and d2y/dx2 of x2 - y2- 2xy = 3 is...
What is dy/dx = 1 and d2y/dx2 = 0