Linearization
Use the linearization of h(x) = √(x2 − 5) at x = 3 to approximate h(3.02).
A. h(3.02) ≈ 1.97
B. h(3.02) ≈ 2.02
C. h(3.02) ≈ 1.98
D. h(3.02) ≈ 2.03
E. h(3.02) ≈ 2.01
D. h(3.02) ≈ 2.03
S25
Suppose that y is a function of x defined implicitly by the equation:
xy = arctan(3y)
Find the value of dy/dx at the point where y = 1/3.
4/9(2-pi)
F21
A child blows a balloon which is assumed to have a perfectly spherical shape. The surface area of the sphere increases at a rate of 12π square inches per second. How fast is the radius of the balloon growing when it reaches 3 inches? Note: The surface area of a sphere with a radius of r is 4πr2.
1/2 inch per second
F23
A polynomial f(x) of the form
f(x) = −x4 + Ax3 + Bx2 + 16x
has a critical point at x = 1, which is also an inflection point. Find the values of the constants A and B.
B = -18
F24
What is the absolute minimum for the function f(x) = 16x3 + 12x2 − 24x on the interval −1 ⩽ x ⩽ 1.
A. 4
B. 0
C. −7
D. −15
E. −20
C. -7
F24
Use linear approximation to estimate f(0.01) where f(x) = 16/[ln(x + 1) + 2]
7.96
S24
Use logarithmic differentiation to compute the derivative of the following function.
y=exp(-1/x2)(sec x)x
Your final answer should be expressed completely in terms of x.
dy/dx = exp(-1/x2)(sec x)x [2/x3 + ln(sec x) + x tan x]
S22
A rectangle has its base on the x-axis and its upper two vertices on the parabola y = 24 − 2x2. What is the largest area the rectangle can have; What are its dimensions?
Area Function: A(x) = 2x(24-2x2) units2
Maximum at x = 2 of 64 units2
Dimensions 4 units, 16 units
S22
Find all critical points of the function f(x) = x2/x-2 where x ≠ 2. Also, classify each critical point as a local maximum, a local minimum, or neither.
x = 0 gives a maximum
x= 4 gives a minimum
F21
Let y be a function of x defined implicitly by the equation y2 + 2y − 2 − ex = 0.
Find dy/dx |(0,1).
A. -1/2
B. 1/4
C. 4
D. 1/2
E. e/2
S25
Find the linearization of f(x) = ex / x + 2 at x = 0.
Use it to estimate f(1).
y = (1/4)x + (1/2)
f(1)≈ 3/4
S23
Let xy - y2 = 8.
a) Find dy/dx.
b) Find an equation for the tangent line at the point (6,4).
a) dy/dx = y/(2y-x)
b) y = 2x - 8
F22
A spherical soap bubble shrinks at a rate of 1cm3/s. How fast is the surface area of the bubble changing when the radius is 2 cm?
Volume of sphere: V = 4/3 (pi)(r2)
Surface area of sphere: SA = 4 (pi)(r2)
-1 cm/sec
F21
Find the absolute maximum and minimum values of the function f(x) = (−x2 + 4x + 1) / (x + 1) on the interval [0, 4].
Absolute Maximum: 2
Absolute Minimum: 1/5
F22
3. Suppose f(x) = (x)x^2. What is f′(x)?
A. (x)x^2(x2(X(x^2)-1))
B. (x)x^2(x+2x ln(x))
C. x2(x(x^2)-1)
D. x + 2x ln(x)
E. x2(x + 2x ln(x))
B. (x)x^2(x+2x ln(x))
F24
Find the linearization of f(x) = (x3 + 1)1/2 at x = 2 and use this to estimate f(2.05).
L(x) = 2x-1
f(2.05) ≈ 3.1
F22
Use logarithmic differentiation to compute the derivative of y = ((2x+1)/x)x. Your answer should be a function of x alone.
dy/dx = ((2x+1)/x)x(ln((2x+1)/x)-1/(2x+1))
S25
A cylindrical can with circular bottom but no lid is to be made from a sheet of metal. What is the largest volume possible, assuming 400cm2 of the metal sheet is used for the sides and bottom? Include units in your answer.
(For a cylinder of height h and radius r, the volume is πr2h and the area around the side plus bottom is 2πrh + πr2.)
8000/3√(3π) cm3
F24
Find the absolute minimum and absolute maximum values of f(x) = −2x3 + 9x2 + 24x on the interval −2 ⩽ x ⩽ 2.
Absolute minimum: -13
Absolute Maximum: 68
F23
Let f(x) = 3x4 − 8x3 + 17. Which of the following statements is true?
A. f has a local maximum at x = 0 and a local minimum at x = 2.
B. f has a local minimum at x = 0 and a local minimum at x = 2.
C. f has no local extremum at x = 0 and f has a local maximum at x = 2.
D. f has no local extremum at x = 0 and f has a local minimum at x = 2.
E. f has no local extrema.
D. f has no local extremum at x = 0 and f has a local minimum at x = 2
S25
Find the linearization of f(x) = √x at x = 16.
L(x) = 1/8(x − 16) + 4 = x/8 + 4
S22
Given y = ln [(x+1)/(x+3)], what is dy/dx?
2/[(x+1)(x+3)]
S25
Jack the gardener wishes to enclose a rectangular shaped garden within a given region, as is shown in the picture. Find the maximum area of the rectangular garden if the boundaries of the given region are the x-axis, the y-axis, and the curve y = −x2 + 12. Be sure to explain why your answer is the maximum area.
16
S24
a) Determine the location of all the critical points of the function f(x) = 4x3-x4-11 and classify them.
b) Determine where f(x) is increasing/decreasing.
c) Determine where f(x) is concave up or down.
a) x=0 (neither),3 (local max)
b) increasing for (-infinity, 3) and decreasing for [3, infinity)
c) concave up for [0,2] and concave down for (-infinity, 0] and [2, infinity)
F19
4. Find d/dx [arctan(x2)].
A. sec2(2x)
B. 2x/(1 + x4)
C. arctan(2x)
D. 2x arctan(x2)
E. 1/(1 + 4x2)
B. 2x/(1 + x4)
F24