AP Calc
The derivative of √x - (1/x(x^1/3) is
(1/2)x^(-1/2) + (4/3)x^(-7/3)
The function f is continuous at x = 1. (√(x+3)-√(3x+1))/(x-1) for x ≠ 1 If f(x) = { k for x = 1 then k =
-(1/2)
Consider the function f defined on pi/2≤x≤3pi/2 by f(x)= tanx/sinx for all x≠pi. If f is continuous at x=pi, then f(pi)=
-1
If y=3/(4+x^2) then dy/dx =
-6x/(4+x^2)^2
if y^2-3x=7 then (d^2y)/(dx^2) =
-9/(4y^3)
If the function f is continuous for all positive real number and if f(x)= (lnx^2 - xlnx)/(x-2) when x≠2, then f(2) =
-ln2
limh->0 ((cos(x+h)-cosx)/h)
-sinx (explanation: definition of a derivative)
limh->0 of (3(1/2+h)^5 - 3(1/2)^5)/h =
15/16 (explanation: def. of a deriv.... f(x)=3x^5)
d/dx[Arctan3x] =
3/(1+9x^2)
If y=u+2e^u and u=1+lnx, find dy/dx when x=(1/e)
3e
If f(x)=e^2x and g(x) = ln(x), then the derivative of y=f(g(x)) at x=e is
2e
The slope of the line tangent to the graph of ln(x+y)=x^2 at the point where x=1 is
2e-1
According to the graph, in what time interval is the speed of the leaf greatest?
3 < t < 5
Let f be a differentiable function such that f(4)=1 and f'(4)=5. If the tangent line to the graph of f at x=4 is used to find an approximation to a zero of f, that approximation is
3.8
If g(x) = (x-1)^(1/3) and f is the inverse function of g, then f'(x) =
3x^2
At how many points on the interval -2pi≤x≤2pi does the tangent to the graph of the curve y=xcosx have slope pi/2?
4
If the line 3x-y+2=0 is tangent in the first quadrant to the curve y=x^3+k, then k=
4
The line x-2y+9=0 is tangent to the graph of y=f(x) at (3,6) and is also parallel to the line through (1,f(1)) and (5,f(5)). If f is differentiable on the closed interval [1,5] and f(10=2, find f(5)
4
Two particles move alone the x-axis and their positions at 0≤t≤2pi are given by x1=cos2t and x2=e^((t-3)/2) - 0.75. For how many values of t do the two particles have the same velocity?
4
The slope of line tangent to the graph of f(x)=ln(e^(2x) + 3sinx) at x=0 is
5
The composite function h is defined by h(x)=f[g(x)], where f and g are functions whose graphs are shown below. The graph of f has horizontal tangents at x=-2 and x=1. The graph of g has horizontal tangents at x=-3,0, and 2. The number of points on the graph of h where there are horizontal tangent lines is
6
If the function f is continuous on the closed interval [a,b] and differentiable on the open interval (a,b) which of the following statements must be true? I. f'(c)= (f(b)-f(a))/(b-a) for at least one number c, a<c<b. II. f'(c)=0 for some number c such that a<c<b III. f has a minimum value on the interval a ≤x≤b
I and III only (explanation: I and III are true bc of the MVT)
Which of the following functions have a derivative at x=0? I. y=|x^3 - 3x^2| II. y=√(x^2+.01) - |x-1| III. y = (e^x)/cosx
I,II,III
The function f is defined on the interval [-4,4] and its graph is shown to the right. Which of the following statements are true? I. limx->1 f(x) = -1 II. limh->0 (f(2+h)-f(2))/h III. limx->-1^+ f(x) = f(-3)
II & III only
The graphs of functions f and g are shown at the right. If h(x) and g[f(x)], which of the following statements are true about the function h? I. h(0)=4 II. h is increasing at x=2 III. The graph of h has a horizontal tangent at x=4
II & III only
The graph of the derivative of a function f is shown to the right. If the graph of f' has horizontal tangents at x=-2 and 1, which of the following is true about the function f? I. f is increasing on the interval (-2,1) II. f is continuous at x=0 III. The graph of f has an inflection point at x=-2
II and III only
The graph of y=(sinx)/x has I. a vertical asymptote at x = 0 II. a horizontal asymptote at y = 0 III. an infinite number of zeros
II and III only (explanation: limx->0 of (sinx)/x = 1)
At x=0, which of the following statements is TRUE of the function f defined by f(x)=√(x^2 + .0001) I. f is discontinuous II. f has a horizontal tangent III. f' is undefined
II only (explanation: I. f(0)=√.0001. III. f' exists)
On which of the following intervals is the graph of the curve y = x^5 - 5x^4 +10x + 15 concave up? I. x < 0 II. 0 < x < 3 III. x > 3
III only
Suppose f is a function whose derivative is given by f'(x) = ((x-1)(x-4)^3)/(1+x^4). Which of the following statements are true? I. The slope of the tangent line to the curve of y = f(x) at x = 2 is -8 II. f is increasing on the interval (1,4) III. f has a local minimum at x = 4
III only
Which of the following is true about the function f if f(x)=(x-1)^2/(2x^2 - 5x +3) ? I. f is continuous at x=1 II. The graph of f has a vertical asymptote at x=1 III. The graph of f has a horizontal asymptote at y=(1/2)
III only (there is a hole at x=1)
If functions f and g are defined so that f'(x)=g'(x) for all real numbers x with f(1)=2 and g(1)=3, then the graph of f and the graph of g
do not intersect (explanation: same slope)
A relative maximum of the function f(x)=(lnx)^2/x occurs at
e^2
Let f be a polynomial function with the values at selected inputs recorded in the table above. Which of the following must be true for -1 < x < 4 ?
f has at least two relative extrema
At x=0, which of the following is true of the function f defined by f(x)=(x^2)/(1+sinx) + e^(-2x) ?
f is decreasing
A particle moves alone a coordinate line so that x, its distance from the origin at time t, t ≥ 0 is given by: x(t)=(cos^3)t. The first time interval in which the point is moving to the right is
pi < t < (3pi/2)
A particle moves along the x-axis so that at time t ≥ 0, its position is given by x(t)=(t+1)(t-3)^3. For what values of t is the velocity of the particle increasing?
t < 1 or t > 3
A particle starts at time t = 0 and moves alone a number line so that its position, at time t ≥ 0, is given by x(t) = (t-2)^3(t-6). The particle is moving to the right for
t > 5
For x ≠ 0, the slope of the tangent to y=xcosx equals zero whenever
tanx=1/x
An equation for a tangent line to the graph of y = Arctan(x/3) at the origin is:
x-3y=0
An equation of the normal to the graph of f(x)=x/(2x-3) at (1,f(1)) is
x-3y=4
Let f(x) = 4x^3 - 3x - 1. An equation of the line tangent to the graph of y = f(x) at x = 2 is
y = 45x - 65
Let f be a differentiable function with f(3)=4 and f'(3)=8, and let g be the function defined by g(x) = x(√(f(x)). Which of the following is an equation of the line tangent to the graph of g at the point where x=3?
y-6=8(x-3)
If g'(x) = 2g(x) and g(-1) = 1, then g(x) = ?
e^(2x+2)
The one-to-one function f is a twice differentiable function with f'(x) > 0 and f''(x) > 0 for all x. If the function g is the inverse of f, g(x) = f^-1(x), the graph of g is
increasing and concave down everywhere
Let m and b be real numbers and let the function f be defined by 1+3bx+2x^2 for x ≤ 1 F(x) = { mx+b for x > 1 If f is both continuous and differentiable at x = 1, then
m = 1 and b = -1
An equation of the line tangent to the graph of y=x^3+3x^2+2 at its point of inflection is
y=-3x+1
Find the point on the graph of y = √x between (1,1) and (9,3) at which the tangent to the graph has the same slope as the line through (1,1) and (9,3).
(4,2)
Consider the curve x+xy+2y^2=6. The slope of the line tangent to the curve at the point (2,1) is
-(1/3)
The graph of y=x^4-x^2-e^(2x) changes concavity at x =
-0.531
What is limx->0 of ((1/(x-1)+1)/x)?
-1
If p(x) = (x+1)(x+k) and if the line tangent to the graph of p at the point (4,p(4)) is parallel to the line 5x-y+6=0, then k =
-2
Functions f and g are defined by f(x)=1/(x^2) and g(x)=arctanx. What is the approximate value of x for which f'(x)=g'(x)?
-2.359
Let y = 2e^(cosx). Both x and y vary with time in such a way that y increases at the constant rate of 5 units per second. The rate at which x is changing when x = pi/2 is
-2.5 units/sec
The function F is defined by F(x)=G[x+G(x)] where the graph of the function G is shown at the right. The approximate value of F'(1) is
-2/3
The slope of the tangent line to the curve 2xy + tiny = 2pi at the point where y = pi is
-2pi
What is limx->∞ of (3x^2 + 1)/(3-x)(3+x) ?
-3 (explanation: exponent on numerator & denominator is equal so the answer is the value of the coefficients on those variables w/ the same exponents)
Suppose the g is a function with the following two properties: g(-x) = g(x) for all x, and g'(a) exists. Which of the following must necessarily be equal to g'(-a)?
-g'(a) (explanation: deriv. of even function = odd function)
If cosx=e^y and 0<x<pi/2, what is dy/dx in terms of x?
-tanx
At what input x do the graph of y=x^2 - 1/(e^x) and y=2(√x) have parallel tangent lines?
0.435
Let f be a function defined by f(x)=(5-2x)(x^(2/3)). f is increasing on the interval
0<x<1
The function f is defined by f(x)=(x-2)^(2/3) + 1. The absolute minimum value of f on the closed interval [1,10] is
1
If g(x)=x+cosx, then limh->0 (g(x+h)-g(x))/h =
1-sinx (explanation: deriv of x+cosx)
What is limx->1 (√x-1)/(x-1) ?
1/2
If g(x) = (x-2)/(x+2), then g'(2) =
1/4
limx->0 (1-cosx)/(2sin^2x) =
1/4
Let f(x) = xlnx. The minimum value attained by f is
1/e
Consider the function f(x)=6x/(a+x^3) for which f'(0) = 3. The value of a is
2
The approximate value of y=√(3+e^x) at x=.08, obtained from the tangent to the graph at x=0 is
2.02
The formula x(t)= lnt + (t^2)/18 +1 gives the position of an object moving alone the x-axis during the time interval 1≤t≤5. At the instant when the acceleration of the object is zero, the velocity is
2/3
The graph of f', the derivative of a function f, is shown at the right. The graph of f' has a horizontal tangent at x=0. Which of the following statements are true about the function f? I. f is increasing on the interval (-2,-1) II. f has an in inflection point at x = 0 III. f is concave up on the interval (-1,0)
I & II only
The function f is continuos and differentiable on the closed interval [1,5]. Which of the following statements must be true?
There exists a number c, 1 < c < 5 for which f(c) = 0 (explanation for right answer: MVT for functions. explanation for wrong answers: we can't determine what is happening between the points)
