High School Calculus Questions

Use the graph of the function to answer each question 2 x 2 f x csc x 2 3 2 a Find all x intercepts of the graph of y f x Enter your answers as a comma separated list If an answer does not exist enter DNE x b Find all y intercepts of the graph of y f x Enter your answers as a comma separated list If an answer does not exist enter DNE y c Find the intervals on which the graph of y f x is increasing and the intervals on which the graph of y f x is decreasing Enter your answers using interval notation If an answer does not exist enter DNE increasing decreasing d Find all relative extrema if any of the graph of y f x Enter your answers as a comma separated list of ordered pairs If an answer does not exist enter DNE relative minima relative maxima e Find all vertical asymptotes if any of the graph of y f x Enter your answers as a comma separated list If an answer does not exist enter DNE

Use the graph of the function to answer each question 2x x 2 f x cot x 2 3 2 a Find all x intercepts of the graph of y f x Enter your answers as a comma separated list If an answer does not exist enter DNE x b Find all y intercepts of the graph of y f x Enter your answers as a comma separated list If an answer does not exist enter DNE y c Find the intervals on which the graph of y f x is increasing and the intervals on which the graph of y f x is decreasing Enter your answers using interval notation If an answer does not exist enter DNE increasing decreasing d Find all relative extrema if any of the graph of y f x Enter your answers as a comma separated list of ordered pairs If an answer does not exist enter DNE relative minima relative maxima e Find all vertical asymptotes if any of the graph of y f x Enter your answers as a comma separated list If an answer does not exist enter DNE

Describe the relationship between the graphs of f and g Consider amplitude period and shifts 4 y 3 2 3 2x J 8 0 g 1 1 k X 2 3 B O The graph of g is a horizontal shift 3 units to the right of the graph of f O The graph of g is a reflection of f in the x axis O The amplitude of g is twice that of the amplitude of f O The graph of g is a vertical shift 3 units upward from the graph of f The period of a is 3 times that of f

Sketch the graph of the y 2 cos x 1 2 37 k k K12 2 k12 3 3 2x 2 3 KIN KIN KIN 2 k KIN 13 X 2 k2 KIN k

Let A x represent the area bounded by the graph the horizontal axis and the vertical lines at t 0 and x for the graph below Evaluate A x for x 1 2 3 and 4 5 3 M 2 1 A 1 A 2 1 A 3 A 4 2 3 50

Approximate the area under the curve graphed below from x approximation with 4 subdivisions 3 to x 7 using a Left En You will need to approximate the function values using the graph 5 4 3 2 1 1 2 3 A 4 5 6 7 Poot 8

A rectangular soccer field has dimensions 70 yards by 110 yards as shown in the figure A sprinkler placed at point P rotates 15 and sprays across the entire field as shown in the figure What percent of the soccer field is watered by thi one sprinkler Round to the nearest hundredth P 95 yd 15 110 yards 70 yards

8 Wayne wants to plant a triangular garden in his backyard and he will enclose the garden with 204 feet of fencing a Give two possible sets of side lengths for the garden one making an acute triangle and one making an obtuse triangle b Find the semiperimeter of each garden c Find the area of each garden d Wayne decides that he will have one side length of the garden be 50 feet If another side length is x write an expression for the 3rd side length e Write an equation in terms of x for the area of the garden f Use technology to find the lengths of the sides of the garden that give the maximum area and state the maximum area

Two planes leave the Atlanta airport at the same time and fly for three hours Flight 1504 flies 34 West of North at 245 miles per hour and Flight 168 flies 70 East of North at 271 miles per hour After 3 hours how far apart are the planes Round to the nearest hundredth 34 N 70

2 The figure shows ABRO B 18 9 R 13 a Find the semi perimeter of ABRO b Find the area of ABRO Round to the nearest tenth

4 AVMN has sides of lengths v n and m a Write an expression for the height of the triangle h in terms of the measures of the sides and angles of the triangle b Write an expression for the area of AVMN V m n N

1 In AACD AC 25 inches and DC 18 inches a Find ED to the nearest tenth b Find the area of AACD D

B Find the area of AMPO Round to the nearest hundredth M 14 66 O P 17 5

y f x R S 10 1 Let f be the function given by f x x 3cos x Let R be the shaded region in the second 2 quadrant bounded by the graph of f and let S be the shaded region bounded by the graph of and line l the line tangent to the graph of at x 0 as shown above a Find the area of R

DUE MONDAY APRIL 15th YOU 2004 AP CALCULUS AB FREE RESPONSE QUESTIONS y g x 2 Let f and g be the functions given by f x 2x 1 x and g x 3 x 1 x for 0 x 1 The graphs of and g are shown in the figure above a Find the area of the shaded region enclosed by the graphs of f and g y f x g x 0 1 Let f and g be the functions given by f x 1 sin 2x and g x e 2 Let R be the shaded region in the

A Solve for the missing angle to 400 50 5 degrees 12 7 degrees 36 9 degrees 72 2 degrees 300

Susan is flying a kite which gets caught in the top of a tree Use the diagram to estimate the height of the tree 63 6 ft 1 59 ft 74 2 ft 86 9 ft string 44 90 feet

Solve for x 23 63 50 7 50 1 60 7 X

The function is given by j x 4 3 tan A x 1 2zk where k is an integer B x k where k is an integer 22 C x zk where k is an integer D x x 2xk where k is an integer Which of the following gives the vertical asymptotes of j

10 Given to the right is the graph of f t Additionally let the function g x be defined to be ff t dt a Where does the graph of g have a relative max b Where is the graph of g concave down

q The graph of the function f in the figure below consists of line segments and a quarter of a circle Let g be the function given by g x f t dt Determine the following a g 6 sidot art mo Dedi ort ofbmixo quo of almidue Strange 0 toq mawa oy bbw b g 4 de quoy al fotonu palabonont auc c g 8 Sydw Seulov U17 90 885 10 93 653211 6 S910

Solve the differential equation with the given conditions 21 f x 12x 6x 1 and f 1 5

5 3 For questions 6 8 use the Fundamental Theorem of Calculus to find the derivative of the functions 6 g s t t2 8 dt 7 F x x 3 t dt 8 g x 3 2t 5 dt

The derivative of a function fis defined by f x g x for 4 x 0 15e x 3 3 for 0 x 4 The graph of the continuous function f shown in the figure below has x intercepts at 3 In 3 The graph of g on 4 x 0 is a semicircle and f 0 5 x 2 and x 0 Graph of f For 4 x 4 find all values of x at which the graph of fhas a point of inflection Justify your answer a b c Find f 4 and f 4 For 4x4 find the value at which fhas an absolute maximum Justify your answer

4 Oil is leaking from a pipeline on the surface of a lake and forms an oil slick whose volume increases at a constant rate of 2000 cubic centimeters per minute The oil slick takes the form of a right circular cylinder with both its radius and height changing with time Note the volume V of a right circular cylinder with radius r and height his given by V r h b A recovery device arrives on the scene and begins removing oil The rate at which oil is removed is R t 400 t cubic centimeters per minute where t is the time in minutes since the device began working Oil continues to leak at the rate of 2000 cubic centimeters per minute Find the time when the oil slick reaches its volume Just

1 Do not use a calculator for this one 5 2 Graph of f 4 4 slope 2 The function fis defined on the closed interval 5 4 The graph of consists of three line segments and is shown in the figure above Let g be the function defined by g x a b f t dt Find g 3 On what open intervals contained in 5 x 4 is the graph of g both increasing and concave down g x The function is defined by h x 9 Find h 3

2 A particle moves along the y axis so that its velocity at time t 0 is given by v t 1 tan et At time t 0 the particle is at y 1 Note tan 1 y arctanx a Find the acceleration of the nesti 1

cos 135 12 cos 2

15 sin 60 16

315 A

Let A x represent the area bounded by the graph the horizontal axis and the vertical lines at t O and t X for the graph below Evaluate A x for x 1 2 3 and 4 5 4 3 2 1 A 1 A 2 A 3 A 4 2 3 5

The function is defined on the closed interval 3 8 and is g graph above Let g be the function defined by g x 71 t de a Find g 3 and g 0 b Does g x have a relat the interval s are g x is concave down Give a reas 1 x 3020 d Find sin 2x Wr Evaluate the following expressions t to the p x x 1 Explain your reason 3x sin 3t 1 dt In 4t dt

3 2 0 1 2 1 4 1 Graph of f 5 2 5 The graph of the function shown above consists of a semicircle and three line segments Let g be the function given by g x t dt a Find g 0 and g 0 b Find all values of x in the open Interval 5 4 at which g attains a relative maximum Justify your answer c Find the absolute minimum value of g on the closed interval 5 4 Justify your answer d Find all values of x in the open Interval 5 4 at which the graph of g has a point of Inflection

6 Co primality Let a b be two integers We call a and b relatively primes or co primes IFF gcd a b 1 Show that the numbers 6k 5 and 7k 6 are relatively prime for every k 1 2 2

For the following find F x F x So COS t2 dt F x 2t 5 dt night La 3x F x Int dt aloupe D S F x t dt 03 x Sandwolfof all to d 1 4 A

2 Determine if the integral is convergent or divergent dx If the integral is convergent evaluate x the integral

4 Find m N Round to the nearest tenth N 8 12 11 S

6 In AWAG mZA 64 WA 16 and GA 9 Solve the triangle by finding all the missing measures of the sides and angles Round to the nearest tenth

Write name date and period Show all sketches and work 1 Determine if APQR is an acute obtuse or right triangle Give a reason for your answer P 10 32 6 44 R Q 8 14

5 In ADEF angle F is an obtuse angle Is the value of d x ex cos F positive or negative Explain

Vectors u 2 cos 30 i sin30 j v 6 cos 225 i sin225 j and w 8 cos 120 i sin120 j are given Use exact values when evaluating sine and cosine Part A Convert the vectors to component form and find 7 u v Show every step of your work 4 points Part B Convert the vectors to component form and use the dot product to determine if u and w are parallel orthogonal or neither Justify your answer 6 points

A spherical snowball is melting in such a way that its radius is decreasing at rate of 0 1 cm min At v rate is the volume of the snowball decreasing when the radius is 15 cm The volume of a sphere of radius r is V 4 3 The volume is decreasing at a rate of cm min

2 In AABC mZB 57 and AB 10 Which measure of segment AC would cause there to be two possible ways to draw the triangle A 6 B 9 C 12 D 15

Use a calculator to evaluate the trigonometric function Round your answer to four decimal places Be sure the calculator is se the correct angle mode cot 117

b 1 1 y sin e cos 0 tan e csc e sec 0 cot e

Find the indefinite integral using substitution for 10z 52 2 10x 272 C 5x 4 C 5 4 10x 5x 4 C

Given f x dr 4 f x dz 6 and 6 and 9 2 de 19 dz dx 3 find the following L1 2 dz 2 135 15 3f x g 2 dr 12g 2 dr 0

Find the point x y on the unit circle that corresponds to the real number t t 6 x y 3 2 1 2

A horse owner has 50 lbs of hay that is 6 protein by weight He adds x lbs of oats that is 12 protein by weig The function y 0 06 50 0 12x 50 x models the percent of protein y in the final mixture of feed What is the y intercept of the function that models the percent of protein in the final mixture The y intercept is choose your answer choose your answer 50 50 0 06

S 1 point The function is defined by f z a sin b a c d for constants a b c and d In the y plane the points 2 2 and 4 4 represent a minimum value and a maximum value respectively on the graph of f What are the values of a and d a type your answer d type your answer