unit 1: limits (summer hw)
2. find lim ((x^4 + 2x^3 - 3x^2)/(x^3 - x^2) x→1 =
4
1. this table gives select values of the differentiable function f. x: 6, 7, 10, 13, 15 f(x): -82, -89, -99, -88, -74 what is the best estimate for f^1(9) we can make based on the table?
(-89 - -99)/(7-10) = -3.33
2. simplify the following rational expression that is equivalent to the original? 11k/(55k^2 + 22k). which values of k make the expression undefined?
1/(5k+2) k = 0 and k = -2/5 when undefined
3. average rate of change of h(x) = 1/(10-x) over the interval 5=<x=<8 =
1/10
4. let h(x) = x^(1/4) h^1(16) =
1/32
3. select the equivalent expression. 3^-5 = ?
1/3^5
2. f(x) = x^11 f^1(x) =
11x^10
1. f(x) = x^2 f^1(x) =
2x
1. lim ((x^3 - 2x^2)/(x^2 - 4) x→2 =
1
3. let g(x) = x^(1/4) g^1(x) =
1/4x^(-3/4)
4. select the equivalent expression. a^-1 = ?
1/a
1. average rate of change of f(x) = x^2 + 5x over the interval [1,5] =
11 *plug in both 1 and 5 for the equation and then use those numbers (they are the y values) in the average rate of change equation
1. the following lists the value of functions g and h, and of their derivatives, g^1 and h^1, for x = -2 x: -2 g(x): 1 h(x): -12 g^1(x): 2 h^1(x): 8 let function F be defined as F(x) = 6g(x) + 2h(x) - 1 F^1(-2) =
28
3. simplify the following rational expression. (14x^2 + 7x)/(14x). which values of x make the expression undefined?
2x+1 / 2 x = 0 when undefined
3. lim ((12x - 60)/(x^2 - 6x + 5)) x→5 =
3
3. the following table lists the value of cantons f and g, and of their derivatives, f^1 and g^1, for x = 4. x: 4 f(x): 16 g(x): 12 g^1(x): 7 f^1(x): 8 let function H be defined as H(x) = 3f(x) - 3g(x) + 2. H^1(4) =
3
4. this table gives select values of the differentiable function h. x: -4, -1, 0, 1, 4 h(x): -26, -15, -32, -39, -35 what is the best estimate for h^1(-2) we can make based on this table?
3.67
2. let g(x) = ^3√(x^4) g^1(x) =
4/3x^(1/3)
1. let f(x) = x^5 + 2x^3 - x^2 f^1(x) =
5x^4 + 6x^2 - 2x
1. let f(x) = √(x^3) f^1(16) =
6
4. lim ((x^3 - 9x)/(x^2 - 3x) x→3 =
6
1. select the equivalent expression. 1/(6^5) = ?
6^-5
3. d/dx [x^6] =
6x^5
2. d/dx [x^6 + 3x^4 + 7x^2] =
6x^5 + 12x^3 +14x
4. y = x^7 dy/dx =
7x^6
1. lim sin(x) x→π/3
= (√3)/2
2. lim ((2x+1)/(-3x+3)) x→4
= -1
2. let h(x) = ((x^2 - 49)/(x+7)) when x is not equal to -7 h is continuous for all real numbers find h(-7)
= -14
2. the following table lists the value of functions f and g, and of their derivatives, f^1 and g^1, for x = 1. x: 1 f(x): -4 g(x): -9 f^1(x): 3 g^1(x): -7 Evaluate d/dx [-2f(x) + 3g(x) - 2] at x = 1
= -27
4. lim h(x) x→-1 *the graph starts in QIII and continues through QIV and QI, with a hole at (-1,-3) and a point at (-1,4)
= -3
3. function h is defined for all real numbers lim h(x) x→-2 graph is similar to a quadratic, just shifted down and left... goes through (-2,-3.8)
= -3.8
1. lim f(x) x→3 *the graph leads from quadrant II and then curves down to quadrant IV with an open circle (or a hole) at (3,-4)
= -4
2. g is defined for all real numbers except x = -4 lim g(x) x→-4 *the graph is similar to a x^3 graph and has a hole at (-4,-5.2)
= -5.2
2. lim g(x) x→-8 x: -8.1, -8.01, -8.001, -8.0001, -7.999, -7.99, -7/9 g(x): -0.14, -0.09, -0.02, -0.01, -0.07, -0.11
= 0
4. lim cos(x) x→0
= 0
4. find lim g(x) x→-3 for g(x) = √(7x + 22)
= 1
3. let g(x) = {((√(x+20) - 4)/(x+4)) for x>=-20, x is not equal to 4; k for x = -4 g is continuous for all x>-20 what is the value of k?
= 1/8
1. let g(x) = (x-5)/(√(x-4) - 1) when x is not equal to 5 g is continuous for all x>4 find g(5)
= 2
4. let f(x) = {((15x)/(x^3 + 3x)) for x is not equal to 0; k for x = 0 f is continuous for all real numbers what is the value of k?
= 5
1. find lim h(x) x→6 for h(x) = √(5x+6)
= 6
3. lim (-3x^3 - 7x + 8) x→0
= 8
1. function g is defined over the real numbers x: 0.9, 0.99, 0.999, 1.001, 1.01, 1.1 g(x): 9.12, 9.75, 9.98, 7.03, 7.14, 7.81 lim g(x) x→1
= DNE
1. h is defined for all real numbers except x = 4 lim h(x) x→4 *the graph has a vertical asymptote at x = 4
= DNE
2. function g is defined over the real numbers x: 4.9, 4.99, 4.999, 5.001, 5.01, 5.1 g(x): 0.3, -1.01, -1.99, 6.01, 6.14, 6.33 lim g(x) x→5
= DNE
3. function h is defined over the real numbers x: 3.9, 3.99, 3.999, 4.001, 4.01, 4.1 h(x): -0.39, -0.17, -0.03, 4.98, 4.78, 4.45 lim h(x) x→4
= DNE
3. lim g(x) x→-4 *left side of the graph starts in quadrant II and then leads down to quadrant III, ending with a closed circle at (-4,-3). the right side of the graph starts with an open circle at (-4,2) and then continues to the right
= DNE
4. function f is defined for all real numbers lim f(x) x→-5 *the left side of the graph starts in QIII and ends with a hole at (-5,-4) and then the right side of the graph starts with a closed circle at (-5,1) and then continues to the right
= DNE
4. function h is defined over the real numbers x: -10.1, -10.01, -10.001, -9.999, -9.99, -9.9 h(x): -9.89, -9.47, -9.02, -8.01, -8.3, -8.94 what is a reasonable estimate for lim h(x) x→10
= DNE the limit does not exist
2. lim tan(x) x→π/2
= undefined the limit DNE
3. lim cot(x) x→π/6
= √3
4. Ji-Hun and Wilma manipulated the rational expression ((4z^2 + 4z)/(-8z^2 + 4z)). Ji-Hun = 4z^2/-8z^2, Wilma = (z+1)/(2z+1) Which student wrote an expression that is equivalent to the original?
Neither Ji-Hun (cannot remove addition (additive terms) only) nor Wilma (missing negative sign)
2. exponential to radical form: z^(3/4) =
^4√(z^3)
2. exponential to radical form: z^(1/6) =
^6√z
4. exponential to radical form: x^(4/7) =
^7√(z^4)
4. exponential to radical form: y^(1/7) =
^7√y
3. rational to exponential form: ^5√c =
c^(1/5)
exponential form
completely written out
2. which derivatives is described by the following expression? lim ((ln(x) - 1)/(x - e)) x→e
derivative is at x = e f^1(e), where f(x) = ln(x)
4. continuous at x=3: g(x) = ln(x-3) f(x) = e^(x-3)
f only because there is an asymptote at x = 0 and ln(3-3) = e^x = 0
power rule
f(x)=x^n, n does not equal 0 f'(x)=nx^(n-1)
derivative =
finding slope/average ROC
1. continuous for all real numbers: g(x) = 2^x f(x) = ln(x)
g only
1. compare. g^1(1) ___ g^1(3) *the graph of y = g(x) looks like the x^3 and there is a point at (1,7) and (3,7) and it is steeper (greater slope) at (3,7)
g^1(1) = 2(slope) g^1(3) = 5(slope) g^1(1) < g^1(3)
3. which derivative is described by the following expression? lim ((2^(3+h) - 2^3)/h) h→0
g^1(3), where g(x) = 2^x
2. estimate g^1(4). *the graph of y = g(x) is all in the bottom part of the graph with what looks like an asymptote at y = 0, with a point at (4,-5) and a slope of -3.2
g^1(4) = -3.2
3. continuous for all real numbers: f(x) = tan(x) h(x) = x^3
h only
4. estimate h^1(-1). *the graph is like a really tight and skinny sin graph with a point at (-1,-5), which is at the bottom of one of the downward loops
h^1(-1) = 0
3. compare. h^1(1) ___ h^1(6) *the graph of y = h(x) is in QI with a point at (1,3) and (6,7) and (1,3) is steeper
h^1(1) = 1 h^1(6) = 1/2 h^1(1) > h^1(6)
4. which of the following is equal to g^1(3) for g(x) = x^2?
lim (((3+h)^2 - 9)/(h)) h→0
1. which derivative is described by the following expression? lim ((sin(x) - 0)/(x-π)) x→π
lim ((f(x)-f(t))/(x-t)) x→t derivative is at x = π because of denominator (x-π) and the limit is for x→π sin(x) - 0 = sin(x) - sin(π), therefore be can conclude that f(x) = sin(x) and therefore the answer is f^1(π), where f(x) = sin(x)
definition of a derivative
lim (f(x+Δx)-f(x))/Δx Δx->0
f is continuous at x = a if and only if....
lim f(x) x→a = f(a)
2. continuous at x = 0 g(x) = cot(x) h(x) = 1/(x^2)
neither g nor h
1. naledi and duru manipulated the rational expression (12n^3 + 48n^2)/(12n^3 + 60n^2) naledi = (n+4)/(n+5), duru = (48n^2)/(60n^2) Which student wrote an expression the is equivalent to the original?
only naledi because you cannot remove only addition, which is what duru did
rational form
with the square root symbol (the radical)
3. radical to exponential form: ^5√(z^4) =
y^(4/5)
1. rational to exponential form: ^9√z = *this is not 9*√z, it is: 9 is in the radical (ninth root)
z^(1/9)
1. radical to exponential form: ^8√(z^7) =
z^(7/8)
2. select the equivalent expression. 1/(z^7) = ?
z^-7
4. the following table lists the value of functions g and h, and of their derivatives, g^1 and h^1, for x = 8. x: 4 g(x): 8 h(x): 14 g^1(x): -1 h^1(x): 1 Evaluate d/dx [2g(x) - 4h(x) + 6] at x = 8
-6
4. simplify the following rational expression. ((28 - 7k^2)/(k^2 - 4)) which values of k make the expression undefined?
-7 k = 2 and k = -2 when undefined
4. let h(x) = 1/x^6 h^1(1) =
-6
3. simplify the following rational expression. ((k^2 - 22k + 121)/(k^2 - 121)) which values of k make the expression undefined?
(k-11)/(k+11) k = 11 and k = -11 when undefined
1. simplify the following rational expression. ((k^2 - 11k + 30)/(9k^2 - 90k + 216) which values of k make the expression undefined?
(k-5)/(9(k-4)) k = 4 and k = 6 when undefined
2. simplify the following rational expression. ((x^2 - 26x + 169)/(1859 - 11x^2)) which values of x make the expression undefined?
(x-13)/(-11(x+13)) x = 13 and x = -13 when undefined
2. this table gives select values of the differentiable function h. x: -17, -15, -13, -11, -10, -8 h(x): -60, -63, -69, -86, -85, -88 what is the best estimate for h^1(9) we can make based on this table?
-1.5
4. slope of secant line that intersects graph of h(x) = √(15-2x) at x = 3 and x = 7 =
-1/2
1. let f(x) = x^-2 f^1(2) =
-1/4
3. let g(x) = 1/(x^10) g^1(x) =
-10x^-11
4. what is the value of d/dx (2x^4 + x^3 + 3x^2) at x = -1?
-11
2. let g(x) = x^-12 g^1(x) =
-12x^-13
4. g(x) = {e^x for -5<x<-1; x/e for -1=<x<0 find lim g(x) x→-1^-
-1^- = 1/e = 1/e
3. this table gives select values of the differentiable function f. x: 11, 13, 14, 15, 17 f(x): -52, -63, -63, -63, -68 what is the best estimate for f^1(16) we can make based on this table?
-2.5
2. h(x) = {1/(2x) for x=<-2; 2^x for -2<x=<0 find lim h(x) x→-2
-2^- = -1/4 -2^+ = 1/4 = DNE
1. h(x) = {5x for x<-2; x^3 - 2 for x>=-2 find lim h(x) x→-2
-2^- = -10 -2^+ = -10 = -10
3. let h(x) = 3x^4 - 6x^3 + 2x^2 Find h^1(-1)
-31
2. slope of secant line that intersects graph of h(x) = 16 - x^2 at x = 1 and x = 4 =
-5
3. f(x) = {sin(x*π) for -8<x<0; x/5 for 0=<x=<10 find lim f(x) x→-5
-5^- = 0 = 0
