AP Calculus
Graphing Calculator Skills You may bring to the examination not more than 2 approved calculators. Calculator memories will not be cleared. You are allowed to bring to the examination calculators containing whatever programs you want. SHOWING WORK: • You are expected to show all work. You also may be asked to use complete sentences to explain your methods or the reasonableness of your answers or to interpret your results. • For results obtained from your calculator, you are required to show the setup (the equation being solved, the derivative, or the definite integral being 2
evaluated)
∫x
2
dx = 5.333
−2
• For solutions found using a calculator other than the four mentioned above, you must show mathematical work. (If you are asked to find a relative minimum value of a function, you are expected to use calculus and show the mathematical steps that lead to the answer. It is not sufficient to graph the function.) • When you are asked to justify your answer, the justification must include mathematical reasons, not merely calculator results. Functions, graphs, line analysis, tables, etc., must all be clearly identified. • A decimal answer must be correct to three decimal places unless otherwise indicated. Do not round values in intermediate steps before a final calculation is made. ◆Plot the graph of a function within an arbitrary viewing window. EX1: For -10 < x < 10, how many points of intersection are there for the graphs of y = ex and y = sin x ? A) 0
B) 1
C) 2
D) 3
E) 4
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AP Calculus
◆Find the zeros of functions (solve equations numerically). EX2: Let f be the function given by f(x) = sin x2. The first positive root for f is A) 0
B)
π
C) 1.772 D) 2.507 2 ◆Numerically calculate the derivative of a function (MATH 8).
E)
π
EX3: f(x) = x(cosx)(lnx)(ex); f ’(1) = A) 0
B) 1.000
C) 1.321
D) 1.406
E) 1.469
EX4: Two particles start at the origin and move along the x-axis. For 0 < t < 4, their respective position functions are given to be x1 = ln(t2) and x2 = (t – 5)2. For what value of t does the acceleration of x1 equal the velocity of x2 ? A) –9.060
B) .470
C) 4.960
D) 5.039
E) none
EX5: Let f ' ( x ) =
ln x − cos x for 1 < x < 6. On what interval is f ‘(x) increasing ? ex
A) (1.481, 4.726)
B) (1, 3.105)
EX6: Let f ' ( x ) =
ln x − cos x for 1 < x < 6. On what interval is f increasing ? ex
A) (1.481, 4.726)
B) (1, 3.105)
C) (3.105, 6)
C) (3.105, 6)
D) (1, 4.726)
D) (1, 4.726)
E) (1, 1.481) U ( 4.726,6)
E) (1, 1.481) U ( 4.726,6)
EX7: If c satisfies the conclusion of the Mean Value Theorem for f(x) = sin -1x on the Interval 0 < x < 1, then c is A) .500
B) .771
C) .785
D) 1.000
E) 1.186
EX8: f(x) = sinx. Use the tangent line at x = 1 to estimate f(1.1). A) 0
B) .540
C) .841
D).895
E) 1.000
◆Numerically calculate the value of a definite integral. (MATH 9). Page 2 of 3
AP Calculus
EX9: The area of the region bounded by the curves y = e-x , y = ln x, and the line x = 1 is A) .042
B).054
C) .096
D) .728
x
EX10: Let g(x) be the function given by
(
E) 1.686
)
g ( x ) = ∫ e t t 2 − 1 dt . Which of the following 0
must be true?
A) I only
I. g is decreasing on (0,1) II. g is decreasing on (1,2) III. g(2) > 0 B) II only
C) III only
D) I and III only
E) I, II, and III
EX11: The average value of the function f(x) = cos(x2) on the closed interval [0,2] is A) .231
B) .461
C) .780
D) .977
E) 1.253
EX12: Let R be the region in the first quadrant enclosed by the graphs y = x4 + 1 and y = x + 16. The volume of the solid generated by revolving R about the x-axis is A) 25.616
B) 80.475
C) 507.539
D) 1594.480
E) 3188.959
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