HEBRON HIGH SCHOOL AP CALCULUS SUMMER MATHEMATICS REVIEW PACKET
AP Calculus AB/BC Mathematics fun for all! Written by Alex O’Brien, updated on 8/29/2014
The purpose of this assignment is to make certain that you have much of the mathematical skills you will need to succeed in an AP calculus class. It is NOT meant to punish or scare you! If you struggle with some of these concepts, you are NOT alone! However, calculus is much easier if you are not trying to learn algebra at the same time! This packet will give you the opportunity to review some mathematics before school starts. You should complete this workbook prior to the first day of class, which will be dedicated to answering your questions over the material covered in this workbook. You will turn in your completed packet on the second day of class (for a grade) and you will have an assessment over this material during the first week of class. If you have calculus AB in the spring semester, you will still be required to complete this packet prior to the first day of class. Your teachers recommend you find a study group to work with. We also recommend you do not complete this too early in the summer, as you may not remember all of it when school starts. Alternatively, do not wait until the last week as cramming this packet will not help your long term learning in calculus. For you to CHECK your work, the answers to the odd problems will be posted on the Hebron web site and Mr. O’Brien’s schoolweb page. Again, discussing and working the problems with your peers is encouraged, but copying someone’s answers is not! There is a difference and at this point in your academic career you should know that. If you cannot handle this amount of honesty, you have no business in AP calculus to begin with. Unless marked calculator active (CA) all problems should be solved without a calculator. All work should be done in the packet, and YOU MUST SHOW ALL WORK AND COMMUNICATE YOUR THOUGHTS CLEARLY TO RECEIVE CREDIT! You may print this packet or complete it electronically using a note taking app on your iPad (Notability, Evernote, etc). Feel free to contact us if you have any questions over the summer. Alex O’Brien Calculus AB/BC
[email protected] @mrobrien314 Room 2525
Ryan Woodward Calculus AB
[email protected] Room 2305
I.
Slope
Δy rise y2 − y1 = = Δx run x2 − x1 Find the slope of the line between the given points or find the value of k so the given coordinates have the given slope. ⎛ 1 ⎞ ⎛ −3 3 ⎞ ⎜ − ,5 ⎟ , ⎜ , ⎟ ⎝ 3 ⎠ ⎝ 4 2⎠ slope m =
1.
2.
( −2,5) , ( −2, −8)
1 3. m = , through the point (1,1) and ( 7, k ) 2 4. m = −5, through the points ( −3, −5) and ( k ,5)
II.
Linear Equations
1.
Write an equation for the line with the given information. All of your equations should be given in point-slope form: y − y1 = m( x − x1 ) Other forms of a linear equation: Standard: Ax + By = C , slope-intercept: y = mx + b 1 m = , through the point (1,1) 2
2. Through the points ( 2, 4) , ( −1, 4)
1 3. Perpendicular to y = − x + 4 through (1,1) 3 4.
Parallel to 3x − 2 y = 7 through (1,3)
5. Parallel to x = 4 through the point (-7, 10)
2
III.
Systems of Equations Solve each system using addition, subtraction, substitution, or elimination. Write your answer in (x,y) form. Do not use matrices to solve the system.
1.
⎧ −2 x + 3 y = 13 ⎪ ⎨ 19 3x − 2 y = − ⎪ ⎩ 2
⎧ 3 x − 2 y = 13 ⎩4 x + 7 y = −31
2. ⎨
⎧4 x + 10 y = 40 ⎩ 6 x + 7 y = 28
3. ⎨
IV.
Composite Functions Unless specified, find f ( g ( x) ) and g o f .
1.
f ( x) = x 2 + 2, g ( x) = x − 1
2.
f ( x) =
1 1 , g ( x) = x+2 x −1
3
3.
f ( x) = ln(2 x − 1), g ( x) =
1 x (e + 1) 2
4. Let f ( x) = 3x3 + 2 x. Find f ( x − 3).
V.
Even/Odd functions. Determine if the following functions are even, odd, or neither. Even: f (− x) = f ( x) , Odd f (− x) = − f ( x) 1.
y = 3x 4 − 2 x 2 + 1
2. y = −3x5 − 2 x
3.
y = sin( x)
4. y = cos( x)
( )
5. y = sin x3
VI.
Domain The Domain of a function is the inputs (commonly the x-values). Places NOT in the domain of a function include x values where the denominator is zero and where there is an even root or log of a negative number. Find the domain of the following functions. Write your answers in interval notation.
1. y =
−1 ( x + 5)( x − 2) 4
2.
f ( x) = 3x − 27
3.
f ( x) = ln (3 − x2 )
4.
g ( x) = x − 3
VII.
Asymptotes Horizontal Asymptote: apply the acronym BOBO-BOTN-EATSDC (Bigger On Bottom, Zero, Bigger On Top None, Exponents Are The Same Divide Coefficients). The Bigger refers to the degree of the numerator (top) and denominator (bottom). Only be concerned with the highest power of x on the top and bottom of your fraction when determining which is bigger. Vertical Asymptote: Set the denominator equal to zero and solve for x. As long as the numerator does NOT equal zero at that same x value, then it is a vertical asymptote. If both the numerator and denominator equal zero at an x-value, then it is called a removable discontinuity. Write the equation for each vertical and horizontal asymptote of the graph of the following functions.
1.
2 x3 + 3x 2 − 1 y= x3 − 1
x2 − 2 x + 1 f ( x) = x3 + 1 3.
x3 + 4 x 2 − 2 x 2. g ( x) = x2 − 1
4.
5
f ( x) =
2x x2 + 1
VIII. Inverse Functions To find an inverse, switch the x’s and y’s, and then solve for y. Find the inverse f −1 ( x) for each function. State the Domain of the inverse.
(
1.
)
f ( x) = 3 x 2 − 1
2. g ( x) = ln( x)
3. g (u ) = e2u +1
4.
f ( x) =
2x +1 x −1
(
)
5. Let f and g be inverses with f (2) = 3, g(2) = 7, and f (7) = 2. Find g f −1 (3) .
IX.
Solving Non-Linear Equations with emphasis on Quadratic Equations. Quadratic equations can be solved by factoring, graphing, or the quadratic formula. Solutions are also called roots and/or zeros. The quadratic formula is:
−b ± b2 − 4ac 2a Solve for the given variable. Give all real solutions. x=
1.
2.
x 2 + 2 x = −1
−4 x − 3x 2 = −15
6
3.
3x3 + 27 x = 0
4.
3x 2 + 4 x = 14 + 4 x
5.
− x2 + 4 x = 4
6.
−4 x = − x 2 + 4
7.
6 x3 − 5 x 2 − 6 x = 0
8.
10 x 11 = 11 10 x
9. 0 =
1 − 1 2 1 2 (2 x) − x + 1 ( ) 6 5
7
X.
Solving Inequalities Solve for the given variable. Give your answers in interval notation. 1.
−4 x − 18 ≥ 2
2.
2 3 ≥ 2x + 3 4x − 3
3.
x2 − 5x + 4 ≥ 0
4. 7 x − 2 ≥ −3 x − 5 ≥ 7 x + 8
XI.
Solving absolute value equations and inequalities Solve for the given variable. If applicable, give your answers in interval notation.
1.
x +1 <3 3
2.
2 − 3x ≥ 3
3.
x−2 = 4
4. 3 3 − 4 y + 2 ≤ 5 8
XII. 1.
Logs, exponents, and exponential equations. Graph y = ln( x) . Then state the domain and range.
2. Graph y = e . Then state the domain and range. x
3. Graph y = 1 + e
x −2
. Then state the domain and range.
4. Graph y = ln(2 x − 4) + 1. Then state the domain and range.
9
Exponent Properties: 1. Multiplying terms with the same base: x a xb = x a +b 2. Dividing terms with the same base:
xa = x a −b b x
3. Raising a power to a power:
(x )
4. Power of a product:
(x
5. Power of a quotient:
⎛ xa yb ⎞ x ac y bc ⎜ 2 ⎟ = 2c z ⎝ z ⎠
6. Negative Exponents:
x−a =
a b
a
= x ab
y b z ) = x ac y bc z c c
c
Logarithm Properties: 1. Converting logs to exponents:
1 1 , xa = −a a x x
log b a = x ⇒ b x = a
2. Adding logs with the same base:
logb x + logb y = logb ( xy)
3. Subtracting logs with the same base:
⎛x⎞ logb x − logb y = logb ⎜ ⎟ ⎝ y⎠
4. Logs with an exponent on the argument: log b x n = n log b x 5. Common Log is base 10
log10 x
6. Natural Log is base e
ln x = log e x
7. Change of base formula:
logb a =
10
log a ln a = log b ln b
Simplify the expression. Your answer should have no negative exponents. 5.
( xyz 2 )3 ( x 2 y 4 z )6
(x y 2
)
−3 5 3
z
=
Simplify the expression. Write your answer with all positive exponents and without radicals (use fractional exponents).
6.
(
4
xy 4 z 2
(
2
)
3
−3 5
x y z
xy 2 z
)
3
=
Solve for the variable. 7. log 4 x = 2
8.
x log 4 2 = 1
9.
log x 3 = 2
10.
log10 − x log100 = 2log1000
11. ln( x + 1) = ln(2 x + 1) + ln( x)
11
12.
ex
2
+ 2 x +1
= 1 + ln(1)
13. ln(3x) = 8ln( x) 4
14. 83 x = 43 x− 2
15.
e2 x − 2e x + 1 = 0
16. (CA) If the population of Beverly Hills high school increases 3% per year, how long will it take for the population of the high school to double (to the nearest tenth of a year)?
17. Simplify to one common log term:
log uv = ln uv
12
XIII. Trigonometry: Radians and degrees and reference angles One radian is defined as the central angle formed from an arc length of one radius length (see below). In other words, if you go the distance of one radius length of a circle around the circumference of the circle, the central angle formed is defined as one radian. There are always 2π radians in a circle. In calculus we use radians 99% of the time. One Radian 2π radians
To convert between radians and degrees, use the following formulas: π 180 o xo = x radians y radians = y 180 π The Unit circle, pictured below, is very important to memorize for success in AP calculus. The unit circle is a circle of radius one with a circumference of 2π . There are 360oin a circle and each of the degree measurements corresponds to a radian measurement. The unit circle has the more “important” angles labeled.
The ( x, y ) coordinates labeled on the circle represent points on the equation x 2 + y 2 = 1 as sin θ well as sine and cosine values given by ( x, y ) = (cos θ ,sin θ ) . *note: tan θ = cos θ 13
To find an angle on the unit circle, always move from the positive x-axis (initial ray) in a counter-clockwise motion. If the angle is negative, you start at the positive x-axis and move in a clockwise direction. If the angle is larger than 360o, or 2π radians, you continue to move around the circle until you reach the angle. For example 405o will go completely around the circle one time, and then an additional 45o . Thus, 405o and 45o are called co-terminal angles because they end in the same terminal ray. Furthermore, −315o is co-terminal to 405o and 45o . You can continually add/subtract 360o or 2π radians to any angle to obtain an infinite number of co-terminal angles. Reference angles always have a value between zero and 90 degrees, inclusive (an angle on the unit circle in the first quadrant). The reference angle is the angle between the terminal ray and the x-axis. Ex: The two angles below are co-terminal and the reference angle is 45o because the angle between the terminal ray shown and the x-axis is 45o .
1.
Label the blank Unit circle below. You can refer to the previous page, but remember, you must memorize this! (No Joke – MEMORIZE this!!)
14
Convert from degrees to radians or radians to degrees. Then state the reference angle in radians. 2.
45o
3.
13π 6
4.
495o
5.
π 2
XIV. Trigonometry: Right Triangle trig Trigonometric functions can be applied to right triangles. A simple mnemonic device is “SOH CAH TOA”, (pronounced “soak uh toe uh”). It stands for: Opposite Adjacent Opposite SOH ⇒ sin(θ ) = CAH ⇒ cos(θ ) = TOA ⇒ tan(θ ) = Hypotenuse Hypotenuse Adjacent 1.
Use the triangle below to find sine, cosine, and tangent of angle B in terms of a, b, and c.
2.
Let a right triangle have sides of lengths 5, 12, and 13. Let θ be the smallest angle. Find the value of all 6 trig functions.
3. If sin θ =
3 π and 0 < θ < , find the value of the other 5 trig functions. 5 2
15
Other important right triangle trigonometric definitions are the following (note they are reciprocals of sine, cosine, and tangent): Hypotenuse Hypotenuse Opposite csc(θ ) = sec(θ ) = cot(θ ) = Opposite Adjacent Adjacent To remember the reciprocals, know that there can be only one “co” in the pair. Cosecant is the reciprocal of Sine, Secant is the reciprocal of Cosine, and Cotangent is the reciprocal of Tangent. These reciprocal identities hold true on more than just right triangles.
XV.
Trigonometry: Circular trig. The sign of a trig function in a specific quadrant in the unit circle can be found using the mnemonic “All Students Take Calculus.” The first letters (ASTC) stand for All, Sine, Tangent, and Cosine. They tell the trig function(s) that are positive in quadrants I through IV respectively. In other words, in quadrant I all trig functions are positive, in quadrant II only sine is positive (and so is cosecant), in quadrant III only tangent is positive (and so is cotangent), and in quadrant IV only cosine is positive (and so is secant). Given A < θ < B , determine the sign of all six trig functions. If the sign cannot be determined (CBD), state as such. The first two are partially completed for you.
1.
9π 3π − <θ < − 8 4
2. −
π 2
<θ <
π 2
5π 3π 3. <θ < 4 2
π 3π 4. <θ < 6 4
sin( x) = CBD
tan( x) =
csc( x) =
cos( x) = neg
cot( x) =
sec( x) =
sin( x) =
tan( x) =
csc( x) =
cos( x) = pos
cot( x) = CBD
sec( x) =
sin( x) =
tan( x) =
csc( x) =
cos( x) =
cot( x) =
sec( x) =
sin( x) =
tan( x) =
csc( x) =
cos( x) =
cot( x) =
sec( x) =
16
XVI. Trigonometry: Circular trig. Continued
y r r sec(θ ) = x
sin(θ ) =
r y y tan(θ ) = x
csc(θ ) =
x r x cot(θ ) = y cos(θ ) =
Let θ be an angle in standard position (initial ray at the positive x-axis) through the given point. Find the value of the given trig function. 1.
( −3,7 )
sec(θ ) =
2.
( −2, −3)
tan(θ ) =
3.
( 4, −3)
csc(θ ) =
4.
( 2, −4)
cot(θ ) =
5.
( −5,12)
sin(θ ) =
6.
( −15, −8)
cos(θ ) = 17
XVII. Trigonometry: Evaluating trigonometric expressions. Evaluate each expression using values from the unit circle and your knowledge of reciprocal identities. Memorize the trig values rather than relying on your calculator or looking back at the unit circle. Note: There may be unit circle values you should know that are not listed here. ⎛π ⎞ 1. sin ⎜ ⎟ = ⎝6⎠
⎛π ⎞ ⎟ − 2sin(π ) = ⎝6⎠
2. 3cos ⎜
3.
2 ⎛ 7π sin ⎜ 3 ⎝ 6 ⎛ 3π ⎝ 4
4. 3csc ⎜
⎞ 1 ⎛π ⎞ − tan ⎟ ⎜ ⎟= ⎠ 2 ⎝4⎠ ⎞ ⎛ 5π ⎟ − csc ⎜ − ⎠ ⎝ 4
⎛π ⎞ ⎛ 11π ⎟ − cot ⎜ ⎝3⎠ ⎝ 4
5. 3sec ⎜
⎞ ⎟= ⎠
⎞ ⎟= ⎠
6. Which of the following are not possible values for secant? 1, e,
XVIII.
2 3 1 2 , , ,π 3 2 π
Inverse Trig functions.
With inverse trig functions you input a value and it outputs an angle. You can convert inverse trigonometric expressions to trigonometric expressions as follows: trig −1 ( x) = y ⇔ trig( y ) = x
⎛π ⎞ ⇔ sin ⎜ ⎟ = 1 2 ⎝2⎠ Inverse trig functions can be represented with the -1 notation or arc notation. cos −1 ( x) = arccos( x) Evaluate the expression or solve the equation involving an inverse trig function. When applicable, give your answer in radians on the interval [0, 2π]. Also, remember the restricted ranges for the inverse trig functions. Ex: sin −1 (1) =
18
π
1.
sin(2 x) =
−1
1 2
⎛ 3⎞
2. sin ⎜⎜ ⎟⎟ = 2 ⎝ ⎠
⎛ 1⎞ ⎟= ⎝ 2⎠
3. arccos ⎜ −
−1
π⎞
⎛ ⎝
⎟= 6⎠
4. tan ⎜ tan
5. 3sin
6.
2
(sin x ) = 27 −1
sec−1 (1) =
XIX. Trigonometry: Graphical behavior of trig functions. The following are the parent function graphs for the 6 trig functions. Note that they are not all drawn on the same scale.
19
A trig function’s period is the interval over which it repeats itself. For sine and cosine, the amplitude is one half of the difference between the maximum and minimum values of the function. The normal period for sine, cosine, secant, and cosecant is 2π , or 360o. The normal period for tangent and cotangent is π , or 180o. A graph of a sine function is given below with the period and amplitude labeled.
In General, for y = a + btrig(cx + d ) , the following applies: 2π d Amplitude = b, Period = , Phase Shift = , Vertical Shift = a c c Only sine and cosine have an amplitude. For tangent and cotangent the period is given by
π
c
.
When applicable, determine the amplitude, period, vertical shift, phase shift, and vertical asymptotes for the given functions. 1.
y = 2sin(π x)
2.
y = 2 − 3cos(2π x + 4π )
20
3.
π⎞ ⎛2 y = −4sin ⎜ x − ⎟ 3⎠ ⎝3
4.
y=
5 ⎛x⎞ tan ⎜ ⎟ 2 ⎝2⎠
Determine the maximum and minimum values of the given functions. 5. y = 2 + 3cos(2 x)
6.
y = −2 + π sin( x)
Describe the difference in the two functions given 7.
y = sin (π x )
y = 4 + sin (π x − 3π )
XX. Trigonometry: Identities Complete the following identities. It may be helpful to refer to a pre-cal book, trigidentities page, or web reference. However, all should be memorized! 1.
sin ( x ) = cos( x)
5.
1 = tan( x)
2.
1 = cos( x)
6.
1 = cot( x)
3.
1 = sin( x)
7.
1 = csc( x)
4.
cos( x) = sin( x)
8.
1 = sec( x)
21
9. sin 2 ( x) + cos2 ( x) =
16. tan(2θ ) =
10. sec2 ( x) − 1 =
17. sin(−θ ) = 18. cos(−θ ) =
11. csc2 ( x) − 1 =
19. tan(−θ ) =
12. tan 2 ( x) + 1 =
20. cos 2 ( x) =
13. 1 + cot ( x) = 2
21. sin 2 ( x) =
14. cos(2θ ) =
22. tan 2 ( x) =
15. sin(2θ ) =
XXI. Trigonometry: Simplifying trig expressions Use identities to simplify the following expressions as much as possible. There may be more than one right answer. 1.
2cos(2 y)sin(2 y) =
tan 2 ( x) − sin 2 ( x) = 2. sec2 ( x)
3.
(sin( x) + cos( x) )(sin( x) − cos( x) ) +1 = sin 2 ( x)
sec2 ( x) csc( x) = 4. csc2 ( x)sec( x)
5. cos( x) tan( x) + cos( x) cot( x) =
22
6.
(sin( x) + cos( x) )
7.
cos4 ( x) − sin 4 ( x) = 1 − tan 4 ( x)
2
− sin(2 x) =
8. 4sin 2 ( x) cos 2 ( x) =
XXII. Trigonometry: Solving equations involving trig. Solve for the variable on the given interval.
sin(2 x) = cos( x) 0 ≤ x ≤ 2π
1.
2. cos(2x) = sin(x) 0 ≤ x ≤ 2π
XXIII.
Right Triangles and Pythagorean’s Theorem
Pythagorean’s theorem: a 2 + b 2 = c 2 c is always the hypotenuse of the right triangle, a and b are the other two sides. Special right triangles: 30 – 60 – 90 45 – 45 – 90 x – x 3 – 2x x–x– x 2 2x x x x 2 x 3 x Answer the questions below about right triangles. 1. Find the legs of the isosceles right triangle with a hypotenuse of length 4.
23
2. In a right triangle, one leg is twice as long as the other. The hypotenuse is 10 units long. Find the area of the triangle.
3.
If Justin walks 8 miles due north and then 10 miles due East, how far is Justin from where he started?
XXIV.
Geometry: Circles
The standard equation for a circle of radius r and center (h, k) is ( x − h)2 + ( y − k )2 = r 2 . 1. Determine the center and radius of a circle with equation ( x − 5)2 + ( y + 9)2 = 25 2. How many times will the circle in number 1 touch the x-axis? The y-axis?
3. Find the area of the circle x 2 − 8x + y 2 + 8 y = 4
Use the diagram below to answer the questions that follow.
4.
In the figure above, each smaller circle is the same size and tangent to the larger circle. If the radius of each smaller circle is 3, find the area of the larger circle.
24
XXV. Geometry: Area, Volume , and Surface Area Answer the following questions about area, volume, and/or surface area of the given shape. You may want to use a formula chart to assist you in memorizing the formulas. 1. Find the volume and surface area of a cylinder with radius of 3in and height of 8in. Omit
2. Find the area of an equilateral triangle with side length s.
3. Use your answer in 2 to find the area of an equilateral triangle with side length 4. Omit
4. Find the height of a trapezoid with area of 12 and bases of 2 and 4.
5. Find the volume of a cone with a height of 6 and a diameter of 8.
6.
Find the surface area of a cylinder if the height is 5 and the volume is 80π.
7. (CA) In a cylindrical prism with volume V, the radius in increased by 15% and the height is decreased by 15%. In terms of V, find the new volume.
8. Find the area of the triangle formed by the x and y axis and the line 8 x + 3 y = 12 .
25
9. The ratio of the radius of circle A to the radius of circle B is 9:1. If the area of a circle A is 81π , find the area of circle B. Omit
XXVI.
Geometry: Area and volume
Answer the questions below on area and/or volume. Use the diagram below to answer the questions that follow. Use the diagram below to answer the questions that follow. Let r = 6 and let the area of the larger circle be 64π .
1.
Find R.
2. Find the area inside the larger circle and outside the smaller circle.
Use the diagram of a square inscribed in a circle below to answer the questions that follow. Assume the area of the square is 16 square units.
3.
Find the radius of the circle.
4. Find the area inside the circle and outside the square.
26
XXVII.
Rates and Rates of Change
Answer the following questions involving rates of change. 1.
If Brandon bikes at 8 miles per hour and Zack bikes at 24 miles per hour, how much of a head start should Zack give Brandon in order for them to reach their destination 4 miles away at the same time?
2. Diane can walk at 5 miles per hour and swim at 3 miles per hour. She needs to reach a boat that is 4 miles down a straight coast and 2 miles off the coast. If she walks 2.5 miles down the coast and then swims in a straight line to the boat, how long does it take her to get to the boat (in minutes)?
3. A bus leaves the station at 6AM and travels round trip to Hawk land and back in 300 minutes. If he continues this trip over and over again at a constant speed, what is the first time after 6PM that the bus will be in the station?
4. Kate and Donna leave Hebron at the same time. If Kate drives due North at 24 mph and Donna drives due East at 7 mph, how far apart are they after 102 minutes?
5. A 6ft tall man is standing against a 15ft tall lamppost when he begins walking away at a rate of 3 feet per second. How long is his shadow after 4 seconds?
6. A 25ft ladder is leaning against a house with its base touching the ground 3 ft from the wall. The ladder begins to slide down the wall so that the base moves at a rate of 2ft per second away from the wall. Find the area of the triangle formed between the ladder, ground, and house two seconds after it begins to slide.
27
XXVIII. Simplifying Algebraic expressions Simplify the following expressions. You should have no complex fractions or negative exponents. 1 1− 2 x 1. 1 1+ x
2.
1 1 − x x−h h
2 ( x + h ) − ( x + h) + 3 − 2 x 2 + x − 3 3. h 2
4.
5.
⎛ x⎞ −y + x⎜− ⎟ ⎝ y⎠ y2
(x
2
1 1 − ⎡1 ⎤ + 1) 2 − x ⎢ ( x 2 + 1) 2 (2 x) ⎥ ⎣2 ⎦ 2 x +1
28
6.
XXIX.
(t
2
1 1 − ⎡1 ⎤ + 2t + 3) 2 ⋅ 6t − 3t 2 ⎢ ( t 2 + 2t + 3) 2 (2t + 2) ⎥ ⎣2 ⎦ 2 2 (t + 2t + 3)
Solving for a specific symbol or variable
Isolate the given variable by moving all terms with that variable to one side of the equation. Then factor out the given variable and divide by its coefficient.
(
)
1. Solve for z: 2 x2 + y 2 ( 2x + 2 yz ) = xz + y
2. Solve for the
dy dy dy dy symbol: 3 y 2 + 2 y − 5 − 2 x = 0 dx dx dx dx
3. Solve for the
dy dy ⎛ dy ⎞ symbol: = cos( xy ) ⎜ x + y ⎟ dx dx ⎝ dx ⎠
4. Solve for the
dy dy dy symbol: x 2 + 2 xy + y 2 + 2 xy = 0 dx dx dx
29
XXX.
Evaluating expressions 1.
2.
dV dr dh in the equation below. = 7, = 2, and h = 2r , find the value of dt dt dt dV 4 ⎛ 2 dh dr ⎞ = ⎜π r + 2π rh ⎟ dt 3 ⎝ dt dt ⎠ Given r = 3,
Given x = 3, y = 4,
dx dy dz in the = 2, = −4, and x 2 + y 2 = z 2 , find the value of dt dt dt
equation below. dx dy dz 2x + 2 y = 2z dt dt dt
3.
4.
dx π dθ in the equation below. = 7 and θ = , find the value of dt 6 dt 1 dx dθ = sin θ ⋅ 2 dt dt Given
⎧2 x + 1 x < 1 Given f ( x) = ⎨ 2 , find the following (assume h > 0 ): ⎩x + 2 x ≥ 1 a. f (−1) b. f (1) c. f (5) d. f (1 + h)
30
