PREPARATION FOR CALCULUS

Transcription

1 PREPARATION FOR CALCULUS WORKSHEETS Second Edition DIVISION OF MATHEMATICS ALFRED UNIVERSITY

2 Contents Real Numbers Worksheet Functions and Graphs Worksheet 5 Polynomials Worksheet. Trigonometry Worksheet 8 Trigonometric Functions Worksheet Eponential and Logarithmic Functions Worksheet 5 Rational Functions Worksheet 9 Limits Worksheet Computing Limits Worksheet 6 Limits at Infinity Worksheet 40 Continuous Functions Worksheet 44 Prepared by: Joseph Petrillo, Alfred University Edited by: Xiuhong Du and Juan Marin, Alfred University Editions: 7/09, /0 References: Calculus (Early Trans.), 8 th ed., Anton/Bivens/Davis, Wiley (005). Wolfram s Math World (

3 Real Numbers Worksheet. Solve each equation for. (a) = 0 Preparation for Calculus Worksheets (b) = 0 4 (c) = 0. Perform the following operations and simplify where possible. (a) = (b) 5 + = 9 6 (c) + =

4 Preparation for Calculus Worksheets (d) + = (e) + + = (f) 0 5 = (g) = 5 (h) h h + =

5 Preparation for Calculus Worksheets. Use the properties of eponents to simplify. Rewrite any negative eponents. (a) ( 8) = (b) (7 ) = (c) 4 = ( ) (d) 4 ( ) = (e) ( + ) =

6 Preparation for Calculus Worksheets 4 4. Solve each equation for. (a) 7 = (b) + = 5 (c) = Calculate or simplify each of the following. (a) n n! 9! (b) = 7! ( n + )! (c) = n! ( n )! (d) = ( n + )! (n + )! (e) = (n)!

7 Preparation for Calculus Worksheets 5 Functions and Graphs Worksheet. (a) Does the graph of a circle in the Cartesian plane represent a function? Eplain. (b) The circle + y = called the unit circle. It is centered at the origin and has radius. Solve this equation for y to show how the unit circle can be epressed as two separate functions. (In general, the equation of a circle of radius r and center at the origin is + y = r.). (a) The function y = = is the square-root function, and the function y = = is the cube-root function. Discuss the domain and sketch the graphs.

8 Preparation for Calculus Worksheets 6 (b) The function y = = is the reciprocal function. Discuss the domain and sketch the graphs of y = = and y = =. r (c) Any function of the form y =, where r is real, is a power function. [The functions from parts (a) and (b) are power functions.] Sketch the graphs of the identity function y =, the squaring function y =, and the cubing function y =. (d) Sketch the graph of the absolute value function y =.

9 Preparation for Calculus Worksheets 7. Let f() = ( )( + ). (a) Find the -intercepts of f. (b) Find the y-intercept of f. (c) Find the intervals of positive and negative of f. 4. (a) Sketch the graphs, and then find the points(s) of intersection, if any, of the lines 4y = 7 and + y =. That is, solve the system of equations 4y = 7 + y = (b) Sketch the graphs, and then find the point(s) of intersection, if any, of the circle + y = 8 and the line y = 4. That is, solve the system of equations + y = 8 y = 4

10 Preparation for Calculus Worksheets 8 5. (a) Write y = as a composition of three functions f, g, and h. (b) Find formulas for the compositions g() = +. f o g and go f given that f() = and 6. Find the difference quotient of each function. (a) f() = 6 (b) f() = 7

11 Preparation for Calculus Worksheets 9 (c) f() = (d) f() = (e) f() = (f) f() = 4 + 9

12 Preparation for Calculus Worksheets 0 7. Sketch the graphs of y =, y = + 4, y = + 4, y = + 4, and y = (a) Sketch the graph of y =. If possible, find a formula for the inverse and sketch its graph on the same set of aes. If the function is not invertible, then restrict its domain so that an inverse can be found. (b) Sketch the graph of y =. If possible, find a formula for the inverse and sketch its graph on the same set of aes. If the function is not invertible, then restrict its domain so that an inverse can be found.

13 Preparation for Calculus Worksheets Functions in applications 9. For a given outside temperature T in degrees Fahrenheit, the wind chill temperature (WCT) inde is the equivalent temperature that eposed skin would feel with a wind speed of v miles per hour. T, if 0 v WCT = T 5.75v Tv, Find the WCT to the nearest degree if T = o 0 F and v = 5 mi/h. if v > 0. The Surface area S and the volume V of a spherical balloon can be viewed as functions of the radius r of the balloon. That is, S( r) 4 r = π and 4 V ( r) r = π. Find the surface area and volume of a spherical balloon with a -inch radius. Eplain your answers in terms of the units involved.

14 Polynomials Worksheet Preparation for Calculus Worksheets. y = is a constant function and has degree 0. It s graph is a horizontal line. (a) Find the y-intercept. (b) Find the -intercept(s).. y = is a linear function and has degree. It s graph is a line with slope and y-intercept. Find the -intercept(s).. A line passes through the points (, 7) and (4, ). (a) Find the slope of the line. (b) Find the point-slope form of the line. (c) Find the slope-intercept form of the line.

15 Preparation for Calculus Worksheets 4. y = is a quadratic function and has degree. It s graph is a parabola. (a) Find the y-intercept. (b) Find the -intercept(s). (The equation factors easily.) (c) Determine the intervals on which the polynomial is positive and the intervals on which the polynomial is negative. 5. y = is a cubic function and has degree. (a) Find all roots. (Use grouping.) (b) Determine the intervals on which the polynomial is positive and the intervals on which the polynomial is negative.

16 Preparation for Calculus Worksheets 4 6. y = is a cubic function. (a) Find all zeros. (Use guess-and-check and long division. The remaining quadratic factors easily.) (b) Determine the intervals on which the polynomial is positive and the intervals on which the polynomial is negative. 7. y = 4 4 is a quartic function and has degree 4. Factor the function using the difference of squares formula.

17 Preparation for Calculus Worksheets y = is a quartic function. Find the roots. (You will eventually need the quadratic formula.) 9. Solve = 0 by completing the square. 0. Complete the square on y = + 4 to determine the verte of the graph.

18 Preparation for Calculus Worksheets 6. Even though some functions are not polynomials, we can use similar techniques. (a) Factor the function f ( ) = 6 +. (Factor out the smallest power of.) 4 (b) Factor the function g ( ) = + 4, and then find its domain and intercepts.

19 Preparation for Calculus Worksheets 7 Polynomials in applications:. Let s ( t) = 6t + 44 be the position in feet of a falling object t seconds after it was dropped. (a) Find the height from which the object was dropped. (b) At what time did the object hit the ground?. Let v ( t) = 9.8t be the velocity in meters per second of a moving object t seconds after it was thrown straight up into the air. (a) What was the initial velocity? (b) At what time did the object reach its maimum height and begin to descend?

20 Trigonometry Worksheet. (a) Convert o 75 and Preparation for Calculus Worksheets 8 o 5 to radians. π 7π (b) Convert and to degrees Fill in the table from memory. θ cos θ sin θ tan θ o 0 = 0 π = 0 o 6 π = 45 o 4 π = 60 o π = 90 o o π = 80 π = 70 5 π 6 = 4 π = 50 o o o 40. Given that tan θ =, find the eact values of the remaining five trigonometric functions of θ. [Hint: draw the appropriate triangle.] 4. Find the cosine, sine, and tangent of θ. (a) (b) 5 7 θ 4 θ

21 Preparation for Calculus Worksheets 9 5. (a) Find all values of θ between 0 and π (in radians) such that 4sin θ = 0. (b) Find all values of θ between 0 and π (in radians) such that sin θ = cos θ. 6. Find the difference quotient of f() = sin.

22 Preparation for Calculus Worksheets 0 Trigonometry in applications: o 7. A 0-foot ladder leans against a house and makes an angle of 60 with level ground. How far is the top of the ladder above the ground? How far is the bottom of the ladder from the base of the house? 8. An airplane flies over a radar station and then a checkpoint mile away, both located on level ground. At the moment the angle of elevation of the airplane above the radar station is 50 and the angle between the station and checkpoint is 0, find the distance between the airplane and the checkpoint using the Law of Sines, and then find the distance between the airplane and the radar station using the Law of Cosines mi

23 Preparation for Calculus Worksheets Trigonometric Functions Worksheet. Find the amplitude, period, frequency, and phase shift. Then sketch a graph showing at least two periods. (a) y = cos(4) π (b) y = sin( + ) + [Notice the etra vertical shift.]

24 Preparation for Calculus Worksheets. (a) cos = (b) sin () = (c) tan () = (d) sin =. Find θ. (a) θ (b) 50 θ 0

25 Preparation for Calculus Worksheets Trigonometric functions in applications: 4. Suppose a mass is attached to a hanging spring and is allowed to come to rest at its equilibrium position. The mass is pulled 0.5 meters below equilibrium and is released at time t = 0. Assume the mass vibrates up and down with position given by t seconds after release. y( t) = 0.5cos(.t ) meters, (a) Find the amplitude, period, and frequency of the vibration. (b) Find the position of the mass after seconds. 5. In the United States, a standard electrical outlet supplies sinusoidal electrical current with a maimum voltage of V = 0 volts (V) at a frequency of 60 hertz (Hz). Write an equation that epresses V as a function of the time t, assuming that V = 0 if t = 0. [Note: Hz = cycle per second.]

26 Preparation for Calculus Worksheets 4 6. A soccer player kicks a ball with an initial speed of 4 m/s at an angle θ with the horizontal see the figure below. The ball lands 8 m down the field. If air resistance is neglected, then the ball will have a parabolic trajectory and the horizontal range R will be given by v R = sin θ g where v is the initial speed of the ball and g = 9.8 m/s is the acceleration due to gravity. Approimate two values of θ, to the nearest degree, at which the ball could have been kicked. θ R

27 Preparation for Calculus Worksheets 5 Eponential and Logarithmic Functions Worksheet. On the set of aes below, sketch and label the graphs of y = b for bases b =,,, and ½.. Find the eact value of each epression without a calculator. (a) ( 8) = (b) 8 = (c) 9 = (d) log = (e) log 0 (0.0) = (f) log = (g) ln e = (h) (ln e ) =

28 Preparation for Calculus Worksheets 6. Most calculators do not have a key to evaluate log 5. Use the change of base formula to convert to base e first. 4. Solve each equation for. (a) log 0 ( + ) = (b) ln(4 ) ln( ) = ln (c) 5 = (d) ep( ) = 7

29 Preparation for Calculus Worksheets 7 Eponential and logarithmic functions in applications: 5. The loudness of a sound can be measured by its intensity I (in watts per square meter), which is related to the energy transmitted by the sound wave the greater the intensity, the greater the transmitted energy, and the louder the sound is perceived by the human ear. Since intensity units vary over an enormous range, we measure loudness in terms of sound level β (in decibels db): I β = 0log db 0 Damage to the average ear occurs at 90 db or greater. Find the decibel level of each of the following sounds and state whether it will cause ear damage. Sound Intensity I (a) Jet aircraft from 50 ft.0 0 W/m (b) Amplified rock music.0 W/m (c) Garbage disposal W/m (d) TV mid volume from 0 ft. 0 5 W/m

30 Preparation for Calculus Worksheets 8 t 6. The equation Q = e gives the mass Q in grams of radioactive potassium-4 that will remain from some initial quantity after t hours of radioactive decay. (a) How many grams were there initially? (b) How many grams remain after 4 hours? (c) What is the half-life of potassium-4. That is, how long will it take to reduce the amount of radioactive potassium-4 to half of the initial amount? Q 7. In thermodynamics, an equation or the form = Aep t RT function of /T, namely, ln t = a + b T is rewritten as a linear where a is the slope and b is the (etrapolated) vertical ais intercept. Find the slope a and intercept b in terms of A, Q, and R.

31 Preparation for Calculus Worksheets 9 Rational Functions Worksheet. For each rational function, find the following, if possible. (i) (ii) y-intercepts -intercepts (iii) holes (iv) vertical asymptotes (v) intervals of positive and negative (a) 5 y = (b) + 4 y = + 9

32 Preparation for Calculus Worksheets 0 (c) y = + (d) y = 9 + 6

33 Preparation for Calculus Worksheets. Perform long division to rewrite the improper rational function + y =. Rational functions in applications:. According to Coulomb s law, the magnitude of the electrical force F between two charged particles with charges Q and Q is inversely proportional to the square of the distance d between them. That is, kqq F =. d Describe the force as the particles get closer and closer together. Describe the force as the particles get further and further apart.

34 Limits Worksheet Preparation for Calculus Worksheets. Use the graph of y = f() to fill in the blanks. y 6 f() (a) f( ) lim f ( ) lim f ( ) lim f ( ) + (b) f() lim f ( ) lim f ( ) lim f ( ) + (c) f() lim f ( ) lim f ( ) lim f ( ) + (d) f(4) lim f ( ) lim f ( ) lim f ( )

35 Preparation for Calculus Worksheets. Use the graph of y = g() to fill in the blanks. y 6 g() (a) g( ) lim g( ) lim g( ) lim g( ) + (b) g() lim g( ) lim g( ) lim g( ) + (c) g() lim g( ) lim g( ) lim g( ) + (d) g(5) lim g( ) lim g( ) lim g( )

36 Preparation for Calculus Worksheets 4. Numerically analyze the following limits. (a) lim From the left side? From the right side (b) lim From the left side? From the right side (c) lim ) ( ( ) From the left side? From the right side (d) sin lim sin From the left side? From the right side

37 Preparation for Calculus Worksheets 5 Limits in applications: 4. In the special theory of relativity the mass m of a moving object is a function m = m(v) of the object s speed v. In the figure, c denotes the speed of light. Mass m m 0 (a) What is the physical interpretation of m 0? c Speed v (b) What is lim m( v)? What is the physical significance of this limit? v c 5. In the special theory of relativity the length l of a narrow rod moving longitudinally is a function l = l(v) of the rod s speed v. In the figure, c denotes the speed of light. Length l l 0 c Speed v (a) What is the physical interpretation of l 0? (b) What is lim l( v)? What is the physical significance of this limit? v c

38 Preparation for Calculus Worksheets 6 Computing Limits Worksheet Compute the following limits.. lim5. lim( + + 5). lim cos π lim + 5. lim lim tan π

39 Preparation for Calculus Worksheets 7 7. lim 4 8 Sign Chart: N D 4 N/D lim + + Sign Chart: N D N/D 9. 4 lim

40 Preparation for Calculus Worksheets lim lim Sign Chart: N D 5 N/D. lim 9 9

41 Preparation for Calculus Worksheets 9. lim f ( t), where t t, t f ( t) =. 4t 5, t > 4. Find the derivative of each function. (a) f ( ) = (b) f ( ) = +

42 Limits at Infinity Worksheet Preparation for Calculus Worksheets 40. Investigate the end behavior of each of the following functions. + (a) y = (b) f ( ) =

43 Preparation for Calculus Worksheets 4 (c) g( ) = + 4 (d) h( ) = + 4

44 Preparation for Calculus Worksheets 4. Find any horizontal asymptotes of + y = Evaluate each limit. (a) + lim + 8 e (b) lim 9 e (c) lim 9

45 Preparation for Calculus Worksheets 4 Limits at infinity in applications: 4. Let T = f(t) denote the temperature of a baked potato t minutes after it has been removed from a hot oven. The accompanying figure shows the temperature versus time curve for the potato, where r is the temperature of the room. Temp ( o F) 400 r T T = f(t) Time (min) t (a) What is the physical significance of lim f ( t)? + t 0 (b) What is the physical significance of lim f ( t)? t + 5. Suppose that the speed v (in ft/s) of a sky diver t seconds after leaping from a plane is given by the equation v(t) = 90 ( e 0.68t ). (a) Show that the graph of v(t) has a horizontal asymptote v = c for some constant c. (b) What is the physical significance of the constant c in part (b)? (c) Sketch a graph showing v versus t. v t

46 Preparation for Calculus Worksheets 44 Continuous Functions Worksheet. Find constants c and d that make the piecewise function f continuous everywhere.,,, ) ( < < + + = d c f. Use the Intermediate Value Theorem to show that ) ( 5 + = f has three roots in the interval [, ].

47 Preparation for Calculus Worksheets 45 sin. (a) Use the Squeezing Theorem to show that lim =. 0 Hint: In the unit circle, consider the areas of the sector and the two triangles determined by the angle (in radians) shown below. Area of large triangle Area of sector Area of small triangle (b) Use part (a) and the identity sin = cos = ( cos )( + cos ) to show that cos lim = 0. 0

48 Preparation for Calculus Worksheets 46 (c) In a previous eample and on a previous worksheet we calculated the difference quotients of sin and cos, respectively, as follows: sin( + h) sin h sinh cosh = cos sin h h cos( + h) cos h cosh sinh = cos sin h h Use the limits from parts (a) and (b) to calculate the derivatives of sin and cos.