AP CALC Midterm Exam 2020

∫(sin(2x) + cos(2x)) dx =

-1/2 cos(2x) + 1/2 sin(2x) +C

if x²y-3x = y³-3, then at the point (-1,2), dy/dx =

-7/11

lim √2x+5 −1÷x+2 x-->2

1

using the substitution u= 2x+1, ∫²₀ √2x+1 dx is equivalent to

1/2 ∫⁵₁ √u du

consider all right circular cylinders for which the sum of the height and circumference is 30 centimeters. What is the radius of the one with maximum volume?

10/π cm

the first derivative of the function f is defined by f'(x)= sin(x³-x) for 0≤x≤2. On what intervals is f inceasing

1≤x≤1.691

The function f given by f(x) =2x³-3x²-12x has a relative minimum at x =

2

The graph of the derivative of a function f is shown in the figure above. The graph has horizontal tangent lines at x=-1, x=1, and x=3. At which of the following values of x does f have a relative maximum? (test 4)

4 only

The table above gives values of the differentiable functions f and g and their derivatives at x=1. If h(x)- (2f(x) + 3)(1+g(x)), then h'(1) =? (problem set 3)

44

the graph of a function f is shown above. Which of the following could be the graph of f', the derivative of f? (test 3)

B

If f is a continuous function on the closed interval [a,b], which of the following must be true?

There is a number c in the closed interval [a,b] such that f (c)≥f(x) for all x in [a,b]

the graph of f', the derivative of the function f, is shown above. On which of the following intervals is f decreasing? (Problem set 3)

[0,2] and [4,6]

if f'(x) = |x-2|, which of the following could be the graph of y=f(x)? (problem set 3)

e.

the figure above shows the graph of f', the derivative of the function f, on the open interval -7<x<7. If f' has four zeros on -7<x<7, how many relative maxima does f have on -7<x<7? (problem set 4)

one

the function f is continuous for -2≤x≤1 and differentiable for -2<x<1. If (-2) =-5 and f(1)=4, which of the following statements could be false?

there exists c, where -2<c<1, such that f'(c)=0

the derivative of the function f is given by f'(x) =x²cos(x²). How many points of inflection does the graph of f have on the open interval (-2,2)?

five

the function f is continuous for -2≤x≤2 and f (-2)=f(2)=0. If there is no c, where -2<c<2, for which f'(c)=0, which of the following statements must be true?

for some k, where -2<k<2, f'(k) does not exist

the function g(x) = x-5 x>2 -5 x=2 5x-13 x<2 is not continuous at x=2 because

lim G(x)≠ G(2) x→2

)the graph of the function f is shown in the figure above. which of the following statements about f is true? (test 1)

lim f(x) =2 x→a

The figure about shows the graph of a function f with a domain 0≤x≤4. Which of the following statements are true? (problem set 1)

lim f(x) exists x→2⁻ lim f(x) exists x→2⁺

let f be the function given by f(x)= (x-1)(x²-4)/ x²-a. For what positive values of a is f continuous for all real numbers x?

none

for t≥0 hours, H is a differentiable function of t that gives the temperature, in degrees Celsius, at an Arctic weather station. Which of the following is the best interpretation of H'(24)?

the rate at which temperature is changing at the end of the 24th hour

let f be the function given by f(x) = x³-6x². The graph of f is concave up when?

x>2

let f be the function given by f(x)= (2x-1)⁵(x+1). Which of the following is an equation for the line tangent to the graph of f at the point where x=1?

y=21x-19

As a graduation present, Jenna received a sports car which she drives very fast but very, very smoothly and safely. She always covers the 53 miles from her apartment in Austin, Texas to her parent's home in New Braunfels in less than 48 minutes. To slow her down, her dad decides to change the speed limit (he has connections). Which one of the speed limits below is the highest her dad can post, but still catch her speeding at some point on her trip?

65 mph

Let f be a function defined on [-1,1] such that f(-1) = f(1). Consider the following properties that f might have: 1. f is continuous on [-1,1], differentiable on (-1,1) 2. f(x)= cos³x 3. f(x)= |sinπx|

1 and 2 only

the function f is differentiable and has values as shown in the table above. Both f and f' are strictly increasing on the interval 0≤x≤5. Which of the following could be the value of f'(3)? (problem set 3)

30

The function f is given by f(x) = ax²+12/x²+b. The figure above shows a portion of the graph of f. Which of the following could be the values of the constants a and b? (problem set 4)

a = 3 b = -4

for which of the following does f(x) exist? (test 1) x→4

1 and 2 only

which of the following functions below satisfy the hypothesis of the MVT? 1. f(x)= 1/x+1 on [0,2] 2. f(x)= x¹/³ on [0,1] 3. f(x)= |x| on [-1,1]

1 and 2 only

if f'(x) = (x-2)(x-3)²(x-4)³, then f has which of the following relative extrema? 1. a relative maximum at x=2 2. a relative minimum at x=3 3. a relative maximum at x=4

1 only

the function f is continuous and differentiable on the closed interval [3,7]. The table above gives selected values of f on this interval. Which of the following statements must be true? (problem set 4) 1. the minimum value of f on [3,7] is 12 2. there exists c, for 3<c<7, such that f'(c)=0 3. f'(x) >0 for 5<x<7

2 only