Limits
lim x/(x-1)
x→1
a. 1
b. does not exist
c. 1/2
d. 0
b. does not exist
What is y' and g'?
y= tan-1(x)
g= sin-1(x)
y' = 1/(1+x2)
g' = 1/(1-x2)
What are the 4 places a graph will fail to have a derivative?
1. Corner
2. Cusp
3. Vertical tangent line
4. Discontinuities (jump, removable, etc.)
Plug this into your calculator to solve.
2
∫(3x+5x2)dx
-1
58.667
The line y=5 is a horizontal asymptote to the graph of which of the following functions?
(A) y=[sin(5x)]/x
(B) y=5x
(C) y=1/(x-5)
(D) y=5x/(1-x)
(E) y=(20x2-x)/(1+4x2)
(E) y=(20x2-x)/(1+4x2)
What is lim 5x2+4x = ?
x→1
9
What is the derivative of this equation?
ea+7x3
a= 7x2+4x
(14x+4)ea + 21x2
a=7x2+4
Let f be the function defined by f(x)=√(|x-2|) for all x. Which of the following statements is true?
a. f is continuous, but not differentiable at x=2
b. f is differentiable at x=2.
c. f is not continuous at x=2.
d. x=2 is a vertical asymptote of the graph of f.
a. f is continuous, but not differentiable at x=2
What are these two basic integrals?
a. ∫sinudu
b. ∫audu
a. -cosu + C
b. au/lna +C
Explain the difference between the Intermediate Value Theorem and the Mean Value Theorem.
The IVT states that a point on the function f(x) exists, whereas the MVT states that a point on the derivative f'(x) exists.
Evaluate this limit:
lim (x2+x-6)/(x2-4)
x→2
a. -1/4
b. 0
c. 1
d. 5/4
d. 5/4
What is the derivative of cos(x)tan(5x2)?
10xcos(x)sec2(5x2) - sin(x)tan(5x2)
f(x)= { [(2x+1)(x-2)/(x-2)] for x≠2
{ k for x=2
Let f be the function defined above. For what value of k is f continuous at x=2
(A) 0
(B) 1
(C) 2
(D) 5
(D) 5
What is the area of the region in the first quadrant bounded by the graph of y=ex/2 and the line x=2
(A) 2e-2
(B) 2e
(C) (e/2)-1
(D) e-1
(A) 2e-2
Approximate the value of the table using a right Riemann sum.
*You may use a calculator to find the exact answer*
0 | 4 | 6 | 8 | 9 |
-------------------
8 |16|20| 3 |11|
121
An invasive species of plant appears in a fruit grove at time t = 0 and begins to spread. The function C defined by C(t)=7.6arctan(0.2t) models the number of acres in the fruit grove affected by the species t weeks after the species appears. It can be shown that C'(t) = 38/(25+t2).
Write a limit expression that describes the end behavior of the rate of change in the number of acres affected by the species. Then evaluate this limit expression.
lim 38/(25+t2) = 0
x→∞
Let V be the volume of a cylinder with height h and radius r, and assume both vary with time. At a certain instant, the height is 6 inches and is increasing at 1in/sec. At the same instant, the radius is 10 inches and is decreasing at 1in/sec. How fast is the volume changing at that instant?
Volume of a cylinder = 𝝅r2h
dV/dt = -20𝝅 in2/sec
Is the function continuous at x=-1? Show your work using a 3 part definition of continuity.
*May use calculator to graph and use table of values*
f(x)=5x2+e12x+2
Yes, the function is continuous at x=-1.
i. f(-1) = 5
ii. lim f(x) = lim f(x)
x→-1+ x→-1-
iii. f(-1) = lim f(x)
x→-1
Write but do not evaluate an integral expression that gives the volume of a solid generated when R is rotated about the horizontal line y=4.
*Plug these functions into your calculator for a picture*
f(x)=2x2-6x+4 g(x)=4cos((1/4𝝿x))
2
V=𝝿∫[(4-f(x))2-(4-g(x))2]dx
0
Let f(x)=(2x+1)3 and let g be the inverse function of f. Given that f(0)=1, what is the value of g′(1)?
(A) 1/6
(B) 1/54
(C) 1/27
(D) -2/27
(A) 1/6
lim (arcsin(a+h) - arcsin(a))/h = 2
h→0
What could be the value of a?
√3/2
find y"
y=4yx
y" = 32y/(1-4x)2
Find a value for a that will make f(x) continuous.
f(x)= { x2-1, x<3
{ 2ax, x≥3
Hint:Make the left- and right-hand limits to match as x--->3 and check the value of f(3)
a=4/3
Solve this integral.
Hint: You might need to use U-Substitution
4
∫(1/(5-3x))dx
2
-ln(7)/3
A square sheet 4 inches on a side is used to make a box by cutting a small square from each corner and bending up the sides. How large a square can be cut from each corner to make the box have a minimum value?
(2/3)" x (2/3)"