Mth 113 Final Outcomes
NAME
After studying, place a checkmark next to those outcomes you feel you understand and/or are proficient with. Place a question mark
next to those outcomes which you feel your skills/understanding is questionable. Turn in with your test.
To be successful in Mth 113 you should be able to …
Prerequisite Material
1.
Solve linear, quadratic and radical equations algebraically. (QF is adequate)
2.
Rewrite an implicit function in explicit form. i.e. F(x,y) = 0 → y = f(x).
3.
Solve (a) f(t) = g(t) by the intersection method, (b) Solve f(t) = 0 by the root method, (c) Solve f(t) = k by tables.
4.
Graph a line from its equation without the aid of a graphing calculator.
5.
Find the equation of a line from (a) graph (any scale), (b) two points, (c) slope & point, (d) point and bearing,
(e) point and parallel or perpendicular reference, (f) scatter plot (regression)
6.
Graph a function in a 'friendly' window without relying on ZoomFit and find its critical points (roots, extrema, y-intercept)
7.
Graph piecewise functions and rewrite piecewise graphs in algebraic format.
8.
Understand function notation/vocabulary in algebraic, graphic and tabular sense. e.g. domain, range, f(x), f(g(x))
9.
Evaluate functions with (a) a change of variable, (b) at a value, (c) with a new expression. f(x) → f(t), f(2), f(a + b)
10.
Give the domain and range of a function from its algebraic, graphic or tabular form.
11.
Compute the average rate of change (avg slope). i.e. m
ˉ =
12.
Transform a function graphically. i.e. y = f(x) vs. y = ± a f(b(± x ± h)) ± k
13.
Perform various operations among functions. e.g. f(x) + g(x), [f(x)][g(x)], f(g(x)), f 2(x), etc.
14.
Identify the independent vs. the dependent variable.
15.
Use a mathematical model given in an algebraic or graphic form to draw conclusions, make predictions and analyze
behavior inherent in the model.
16.
Switch between the key quadratic forms: y = ax2 + bx + c ↔ y = a(x − h)2 + k ↔ y = a(x − r1)(x − r2)
17.
Find the equation of a quadratic from:
(a) two roots and a third point, (b) vertex and a third point, (c) three random points (regression OK)
18.
Convert angles among various formats: (a) ±θ in DMS, (b) ±θ in radians, (c) bearing, (d) azimuth.
19.
Apply sin θ, cos θ, tan θ, Pythagorean Thm, similar triangles to find missing dimensions numerically.
20.
Use sin-1 y/r, cos-1 x/r, tan-1 y/x in conjunction with a point's quadrant to find the standard angle.
f(b) − f(a)
f(x + h) − f(x)
or
b−a
h
Polar Coordinates
1.
Convert between (x, y) ↔ (r, θ)
2.
Convert between y = f(x) and r = g(θ)
3.
Determine if a polar equation represents a polar function.
4.
Graph polar functions in a friendly window.
5.
Be able to create a specified design/picture using polar functions.
Vectors
1.
Understand the fundamental difference between a vector and a scalar.
2.
Switch between various forms of a vector: (a) descriptive form, (b) graphic form, (c) component form, (d) i, j form
3.
Perform vector arithmetic.
4.
Understand the geometric interpretation of various vector computations.
5.
Use vectors to find resultant components and projections.
6.
Find the angle between two vectors.
7.
Apply vectors to solve applications involving forces and velocities.
u ± v , u • v , u × v , k v , v /k, - v , v̂ , || v ||, etc.
Conic Sections
1. Sketch the graph of as conic section without the aid of a calculator.
2.
Give the equation of a conic section from its graph or other information.
Parametric Functions
1. Sketch the graph of some basic parametric functions without the aid of a calculator.
2.
Sketch the graph of any parametric function with the aid of a calculator.
3.
Modify the parameter's domain to change the range of a parametric function.
4.
Modify the parameter to change the fashion in which the parametric function is plotted.
5.
Give the parametric equation of a parametric function from its graph or other information.
Systems of Equations
1. Solve systems of equations using the addition, substitution or graphing methods.
2.
Use matrices to solve systems of linear equations. Using calculator OK.
1)
Outline the procedure for graphically solving general single variable equations (include both cases f(x) = 0 and f(x) = g(x)).
2)
Give an interpretation of each parameter in y = a f(b(x ± h)) ± k
3)
Algebraically solve for x (check your answer graphically):
(a)
2x + 1
− 2 = 2x + 3
5
(b) 1+
x
x
1−
1+x
= 3x
(c)
2y + 1
= y+1
x−1
(d)
ax2 + 4x + 1 − 2 = x
x2 + 5x + 1 − 2 = 2x − 3
4)
Solve for x algebraically and check your solution graphically:
5)
Solve for y: (a)
6)
Solve for x: (a) 3ax2 + 5bx + 6c = 3x2 + 2x − 5
7)
Solve for x & y: (a) 3x + 7y = -1 & 2y − 4x = 24
8)
Outline the procedure for algebraically finding the equation of a line through two points (a, b) & (c, d).
9)
Find the equation of the line through (a) (5, -7) and (-2, 3), (b) (a, f(a)) and (b, f(b)).
10)
Find the equation of the line through (-3, 8) and (a) parallel to 2x + 5y = 10, (b) perpendicular to 2x + 5y = 10.
11)
Algebraically find the intersection(s) between 2x + 3y = 12 and y = -2x2 + 5x + 7. Then graph and check your answer.
12)
Find a parabola with (a) y-intercept of 4 and vertex at (4, 2), (b) max value of 4 and having roots at x = -3 and x = 7.
13)
For f(x) = 3x2 − 4x + 5, compute and simplify:
14)
When is a polar equation a function? For instance, is the circle r = 4 sin θ a function? Be sure you can justify your answer!
15)
Convert: (a) (-55, 48)xy → (r, θ)
16)
Given:
2y + 1
= y+1
x−1
(b)
2y + 3x
+1 = y+x
2x − 4
(b) (x − a)(x − b) = (x − c)(x − d)
(b) Compute:
(b) ax + by = 1 & x − y = 1
f(x + h) − f(x)
h
(b) (7.2, 4π/3)rθ → (x, y) (c) r2 = cos2 θ − sin2 θ → F(x,y) = 0 (d) xy = 1 → r = f(θ)
u , v , u + v + w , u − v + w , û
u + v , u − v , v̂ , || u ||, u • v , u × v
(c) Compute: 2 u • 3 v , 7.69263434468 w × 3,452,875 w
(d) Show that
18)
1
1
=c+
x−b
x−d
u = (6, -5), v = (-8, 6), w = (10, 12)
(a) Draw and label:
17)
(c) a +
u  w but u is not perpendicular to v
(e) Find compv u
(f) Find proju v
u +v + w+ p = 0
(g) Find p such that
(h) Find q̂ such that q̂ 
One bulldozer pulls a shack as shown. How should the other bulldozer pull to move
the shack due east?
A 1,000# weight (W) is attached as shown. a = 50° and b = 20°. What is the
tension in cable A and cable B? What tension should be applied to cables L and
R to support and balance the system?
v
00
2,0
lbs
Shack
a
A
B
b
L
R
W
19)
20)
A smuggler is headed straight for the landing strip at 120 mph on a heading of N 40° E. At exactly 12 midnight he is 20 miles
from the drop zone. At that point a strong wind begins that has the effect of moving him at 30 mph in direction S 10° E. How
close will he get to the landing zone?
A 150# cat burglar is attempting to access a penthouse as shown.
What is the minimum tension needed in the rope to avoid the tree
if the burglar's feet will hang 5' below the rope.
1)
Case f(x) = 0:
(a) set y1 = f(x), (b) graph, (c) adjust window to see roots, (d) use 2nd CALC to find zeros. x-values of roots are solutions.
Case f(x) = g(x)
(a) set y1 = f(x), y2 = g(x), (b) graph, (c) adjust window to see intersections, (d) use 2nd CALC to find c. x-values of
intersections are solutions.
2)
y=f(x) + k shifts up k, y = f(x) − k shifts down k, y = f(x + h) shifts left h, y = f(x − h) shifts right h, y = a f(x) scales the y-axis
by a, a > 1 stretches, 0 < a < 1 compresses, a < 0 reflects across the x-axis, y = f(bx) scales the x-axis, b > 1 compresses, 0 < a <
1 stretches, b < 0 reflects across the y-axis.
3)
(a) x = -3;
4)
x=3
5)
(a) y =
6)
(a) (3a − 3)x2 + (5b − 2) x + (6c + 5) = 0 Use QF;
x–2
;
3− x
(b) x = 1;
(b) y =
(c) x =
3y + 2
;
y +1
(d) x = ±
3
a−1
2x2 − 9x + 4
6 − 2x
(b) x =
a+b−c−d
;
ab − cd
(c) (a − c)x2 + (-ad − ab + cd + cb − 1)x + (abd +cbd + b) = 0 Use QF.
(b) (x, y) = (
1+b 1−a
,
)
a+b a+b
7)
(a) (x, y) = (2, -1)
8)
(a) Find the slope: use m =
9)
(a) y = -10x/7 + 1/7
(b) m =
10)
(a) y = -2x/5 − 5
(b) y = 5x/2 − 39/2
11)
x=
12)
(a) (⅛)(x − 4)2 + 2
13)
m
ˉ = 6x + 3h − 4
14)
If there is a unique r-value for each θ-value then r = f(θ) is a function. i.e. f(θ1) = f(θ2) whenever θ1 = θ2.
15)
(a) ≈ (73, 2.424)
16)
(b)
b−d
, (b) Find the y-intercept: b = y − mx, (c) Write y = mx + b.
a−c
f(b) − f(a)
, B = f(a) − ma, y = mx + B
b−a
17 ± 505
12
(b) (-4/25)(x + 3)(x − 7)
(b) ≈ (-3.6, -6.235)
(c) (x2 + y2)2 = x2 − y2
(d) r = 1/ sin θ cos θ
u+
v+
w
u • v = -78; u × v = ( 0, 0, -4)
w
u + v = (-2, 1); u − v = (14, -11); v̂ = (-4/5, 3/5); || u || = 61 ;
(c) 2 u • 3 v = -468; 7.69263434468 w × 3,452,875 w = (0, 0, 0)
(d)
u • w = 0 hence perpendicular; u • v ≠ 0 hence not perpendicular
u
w
(e) -39/5
-v
(f) (-78/100)(-8, 6) = (156/25, -117/25)
(g) p = (-8, -13)
v
(h) q̂ = (-2/3, -8/9) or q̂ = (2/3, 8/9)
17)
18)
19)
20)