2019

AP®

ABFragen Free Response Calculation

©2019TheCollegeBoard.CollegeBoard,AdvancedPlacement,AP,APCentralyellogotipoAcornsonmarcascomercialesregistradasdeCollegeBoard.VisitCollegeBoardontheweb:collegeboard.org.

AP Central is the official online home page for the AP program: apcentral.collegeboard.org.

2019AP®CALCULUSABFREERESPONSEFRAGEN

AB CALCULATION

SECTION II, Part A

Time – 30 minutes

Number of questions – 2

A GRAPHING CALCULATOR IS REQUIRED FOR THESE QUESTIONS.

1. Fish enter a lake with a velocity modeled by the function E given by p t

E t( ) = 20 + 15 sin( 6 ). The fish leave the lake at a speed modeled by the function L given by L t 4 20.1 t

2( ) = +

. Both E t( ) and ( )L t are measured in fish by

hour, and t is measured in hours since midnight (t=0).

(a) How many fish enter the lake in the 5-hour period from midnight (t = 0) to 5 a.m.? m.? (t=5) ? Give your answer to the nearest whole number.

(b) How many fish leave the lake on average per hour over a 5-hour period from midnight (t = 0) to 5 a.m.? m.? (t=5)?

(c) At what time t, for 0t £8, is the largest number of fish in the lake? Justify your answer.

(d) Does the rate of change in the number of fish in the lake increase or decrease at 5 a.m.? m.? (t=5) ? Explain your reasoning.

-2- GO TO THE NEXT PAGE.

© 2019 The University Council. Visit the College Board at web:collegeboard.org.

Elt vt 153.457 115 FISH con pritt

iAHY8Yg ut 6.059 Tel B5 O

END.liowTSrNETRtTR ECtLltlf Fttt swce THE NITRATECHANGESSlow

ECHyet 0 NO61203 8 DE POS A NEG AND fb203

kritisch schade LCt THECantidad de FISHCHARMED fromNTf 6.203564 fastC Inc To DEC Besitzer AT E 6.203Let A 6.2035640 fiqh Llthdt0 Maximum AT

f Ect alt dt 135.014 t A1.20318 Ect alt dt 80.919

Elt Ltd is the EXCHANGE RATE in the Number of FISH

Eyes L 5 I

FiEkt LCH it DEKAT 5

2019AP®CALCULUSABFREERESPONSEFRAGEN

2. The speed of a particle P moving along the x-axis is given by the differentiable function Pv, where P v t() is measured in meters per hour and t is measured in hours. The selected values of vt( )P are shown in the table above. The particle P is at the origin at time t = 0.

(a) Explain why there must be at least one time t for 0.3 t £ £2.8 where v ¢(P t), the acceleration of the particle P, is 0 meters per hour per hour.

(b) Use a trapezoidal sum with the three subintervals [0, 0.3], [0.3, 1.7], and [1.7, 2.8] to approximate the

value of v t dt2.8

∫0 PAG( ) .

(c) A second particle, Q, is also moving along the x-axis, so its velocity at 0 t £ £ 4 is given by

Q ( )t cos 0.063t2 v t( ) = 45 meters per hour. Find the time interval in which the velocity Q of the particle

is at least 60 meters per hour. Find the distance traveled by the particle Q during the interval in which the

The velocity of the Q particle is at least 60 meters per hour.

(d) At time t = 0, particle Q is at position x = −90. Use the result from part (b) and the function vQ from part (c) to approximate the distance between particles P and Q at time t = 2.8.

END OF PART A OF SECTION II

-3- GO TO THE NEXT PAGE.

© 2019 The University Council. Visit the College Board at web:collegeboard.org.

I i i

uGAIffdulce EL T.ua III9aYiHzoaPftmEa

ff utdt 3107 HE 2921 40.75 refers to pantosY Y

it tastes delicious

Valet 60 auf AB Att Distance FIVoiLtYdt O6O8uefnSwTEfIYFagiA3.519174 13 PARTC

No 90xofz.no _xCo tf vatlot 45.93710 ABSTAND B w P Q a f2.8

Xp 28 40.75Xabi X 2 8 1 5.187Mangueras

F

2019AP®CALCULUSABFREERESPONSEFRAGEN

AB CALCULATION

SECTION II, Part B

Time: 1 hour

Number of questions – 4

NO COMPUTERS ARE ALLOWED FOR THESE QUESTIONS.

3. The continuous function f is defined on the closed interval −6 £ x 5£. The figure above shows part of the graph of f consisting of two line segments and a quadrant centered around the point (5, 3). We know that the point (3, 3 − 5 ) lies on the graph of f.

(a) Si 5

f x d = 7( ) x ∫−6 , find the value of −2

F X d( )∫ X−6 . Show the work that leads to your answer.

(b) Score 5

∫ (2f x 4 x¢( ) + ) d3 .

(c) The function g is given by −

gramo X( ) = F t re( ) t∫x

two . Find the absolute maximum value of g on the interval

2x − £5. Justify your answer.

(d) Halle ( ) ¢x 10 − 3f x

lim x→1 f x( ) − arctan x

.

-4- GO TO THE NEXT PAGE.

© 2019 The University Council. Visit the College Board at web:collegeboard.org.

μAo ES

fueraAo Ee't

Hμ if he Aaaa

follow

f F4 4 da 21ft x tf 4dxe TBfifthdx Effs ft

2 3 8 8 2 8

G rs 3tBget 0 gts1554104 11 Si un EuspoutsIf fwxufffhddxtfafl.HNgyµ o felices fashion costumes

T.fi oxtTtFgEiThetT.ogaaEMAXTiffani iI 9Ique f q

2019AP®CALCULUSABFREERESPONSEFRAGEN

4. A 2-foot-diameter cylindrical barrel contains rainwater collected as shown in the image above. That

The water drains through a valve (not shown) at the bottom of the barrel. The rate of change of the height h of

the water in the barrel with respect to time t is modeled by dh1

= − hdt 10

, where h is measured in feet and

t is measured in seconds. (The volume V of a cylinder of radius r and height h is V = pr 2h.)

(a) Find the rate of change of the volume of water in the barrel with respect to time when the height of the water is 4 feet. Specify the units of measure.

(b) If the height of the water is 3 feet, does the rate of change of the height of the water increase or decrease with time? Explain your reasoning.

(c) At time t = 0 seconds, the water level is 5 feet. Use variable spacing to find an expression for h in terms of frequency.

-5- GO TO THE NEXT PAGE.

© 2019 The University Council. Visit the College Board at web:collegeboard.org.

Year h rt in h g for tht.ofTinhornr E d I isgne.o.roeodoYh stoafi

ToBof

1stfg omTs Htt parte 4n tooi From it 1h10

to su torn do zfu tot t2F SEPARATION of YANABLESdh Todt Tn tt2Banti Permittitas

tcComstartAND

flitch todtZ

use initial CONDITION

we have sucked Fac hZFS

2019AP®CALCULUSABFREERESPONSEFRAGEN

5. Let R be the area enclosed by the graphs g x( ) = −2 3+cos(p x 2 ) and 2h x( ) = 6 2− ( −x 1) , the y-axis and the vertical line x = become 2 as shown in the figure above.

(a) Find the area of R.

(b) Region R is the base of a solid. For the solid, each x has cross section perpendicular to the x axis

area 1

In x( ) = x + 3

. Find the volume of the solid.

(c) Write, but do not evaluate, an integral expression that gives the volume of the body that results when R rotates about the horizontal line y = 6.

-6- GO TO THE NEXT PAGE.

© 2019 The University Council. Visit the College Board at web:collegeboard.org.

121

yo

4 132x 2 1 zw

121 x gtx d 12 146 2615 C2 3 Ex dX 22 4 2 tarea

R 2xZt43Ex dX zx 9cSNCjIIIETmi r fox zxstai fpswE.ATnorth

1.2 1614 3652625 baby 07 12 f 8 20 tf

Younkefinan buey vol. 304PARTE

AKE AH 3 hktsBohk hls

YIIIEEEEY.if Elri riIoxQoIEEin constant time x2

2019AP®CALCULUSABFREERESPONSEFRAGEN

6. The functions f, g and h are doubly differentiable functions with ( ) 2 g 2 = h( ) = 4. Line 2

y = + ( −x 4 2 3

) es

Tangent to both the graph of g at x = 2 and the graph of h at x = 2.

(a) Hall h¢(2).

(b) Let a be the function given by a x = 3x h x( ) 3 ( ). Write an expression for a (¢ x). Find ¢(a 2).

(c) The function h satisfies 2 x − 4

h x( ) = 1 f x − ( ( ))3

for x π 2. If you know that lim h x(→2 x

) can be evaluated with

L'Hospital's Rule. Use lim h x(x→2

) to find f (2 ) and f ¢(2 ). Show the work that leads to your answers.

(d) It is known that g x h x(( ) £ ) for 1 < 3

## FAQs

### Is AP calc the hardest math class? ›

Finally, **AP Calculus BC is the toughest AP math exam**, if not one of the hardest AP exams period. AP Calculus AB is also challenging, but covers less material and moves more slowly.

**Is AP Calc AB very hard? ›**

In 2021, about 71% of test-takers passed AP exams with a score of 3 or higher. AP Calculus AB students had a pass rate around 61%. Because the pass rate of the AP Calculus AB exam is lower than the average pass rate across all AP classes, you could infer AP Calculus AB is **more difficult than some others**.

**Is it hard to get a 5 on AP Calc AB? ›**

At first glance, it may appear that **it is easier to get a 5 on the AP Calculus BC exam than on the AP Calculus AB exam**. In 2021, for example, about 38% of Calculus BC test-takers received a 5 on the exam, whereas only 18% of Calculus AB test-takers received a 5 that same year.

**What is the most failed AP exam? ›**

The most failed AP exams are **Physics 1 (failed by 48.4% of all students), Environmental Science (failed by 46.6% of all students), and Chemistry (failed by 43.9% of all students)**. For a full chart of the hardest AP exams (those with the lowest passing rate), check out this site.

**What math is higher than Calc? ›**

After completing Calculus I and II, you may continue to Calculus III, **Linear Algebra**, and Differential Equations. These three may be taken in any order that fits your schedule, but the listed order is most common.

**Which AP Calc is harder? ›**

**AP Calculus BC** is one of two AP Calculus courses offered by the College Board, along with AP Calculus AB. AP Calculus BC is the more advanced of the two courses and is designed to build off the skills students learned in AP Calculus AB.

**What is the pass rate for AP calculus? ›**

Exam | 5 | 3 |
---|---|---|

AP Calculus AB | 20.4% | 19.1% |

AP Calculus BC | 41.2% | 20.1% |

AP Computer Science A | 27.3% | 19.9% |

AP Computer Science Principles | 11.4% | 31.1% |

**What is the hardest AP Calc exam? ›**

The **AP Calculus AB exam** is historically one of the hardest AP exams to pass. Its passing rate may look high at 58%, but that's because it's one of the less popular AP exams with a smaller self-selected group of students taking the exam.

**Which math class is hardest? ›**

What is the Hardest Math Class in High School? In most cases, you'll find that **AP Calculus BC or IB Math HL** is the most difficult math course your school offers. Note that AP Calculus BC covers the material in AP Calculus AB but also continues the curriculum, addressing more challenging and advanced concepts.

**What is the hardest math class in the world? ›**

Math 55 is a two-semester long freshman undergraduate mathematics course at Harvard University founded by Lynn Loomis and Shlomo Sternberg. The official titles of the course are Studies in Algebra and Group Theory (Math 55a) and Studies in Real and Complex Analysis (Math 55b).

### Which AP class is the hardest? ›

**The Three Hardest AP Classes**

- AP Physics 1. Despite a reputation as one of the most difficult AP classes, Physics 1 is also one of the most popular—144,526 students took it in 2022. ...
- AP U.S. History. AP U.S. History is one of the hardest AP classes in the humanities and in general. ...
- AP United States Government and Politics.

**What is considered the most difficult math? ›**

**The Riemann Hypothesis**, famously called the holy grail of mathematics, is considered to be one of the toughest problems in all of mathematics.