WorksheetsHonors Calc. - Semester Exam Review
Total questions: 25
Worksheet time: 6hrs 15mins
Find all critical points for f.
f (x) = x2 + 6x - 7
x = 3
x = 2
x = -3
x = -3
Find all critical points for f.
f (x) = x3 + 6x2 - 36x + 5
x = 6, x = -2
x = 4, x = -3
x = -6, x = 2
x = -4, x = -3
Find all critical points for f.
f (x) = x3 + x + 4
x = -1
x = 1
x = -3
No critical points
Find all critical points for f.
f (x) = 1 + 1/x
x = 0
No critical points
x = -1
x = 1
Find all critical points for f.
f (x) = cos(x) - x
x = 1
x = 3π/2 + 2πn
x = π/2 + 2πn
x = π/4 + πn
Give all intervals where the function is concave up.
f (x) = x3 − 6x2 − 36x + 5
(−∞, 2)
(2, ∞)
(−∞, −2)
(−2, ∞)
Give an interval where the function is increasing.
f (x) = x − 1/x
For all real x
x < 0
x > 0
x > 0 and x < 0
True or false, a local extremum can occur at an inflection point.
True
False
True or false, The only polynomials without inflection points have degree 2 or less (i.e., are quadratic, linear or constant functions)
True
False
True or false, if x is a critical point where f'(x) = 0, and if f(x) is a local maximum, then f''(x), if it exists, must be negative.
True
False
f (x) = 1/x2
Find the 10th derivative of f.
47,9001,600x-12
-47,9001,600x-12
39,916,800x-12
-39,916,800x-12
Select an example of a non-linear function which is always increasing
e-x
ln(-x)
-ex
lnx
What is the slope of the tangent line to y3 − x2y = 10 at (3, −2)?
1
-2
3
-4
f (x) = (x + 1)/x
Find the 10th derivative of f.
39,916,800x-11
-39,916,800x-11
3,628,800x-11
-3,628,800x-11
Suppose F and G are two functions which are continuous and differentiable, and such that for all x, F'(x) = G'(x). What do we know about F and G?
F(x) = G(x)
F''(x) = G''(x)
F''(x) = G''(x) and F(x) = G(x)
F and G have the same zeros
Suppose a function is defined and twice differentiable for all x, and suppose it is concave up everywhere. Then True or False, there must be a local minimum value for some value of x.
True
False
Find the maximum and minimum for f on the given interval.
f (x) = 3x3 + x on [a, b]
Minimum: 9a2+ 1, Maximum: 9b2+ 1
Minimum: a, Maximum: b
Minimum: b, Maximum: a
Minimum: 3a3 + a, Maximum: 3b3 + b
Find the absolute maximum and minimum for f on its domain.
f (x) = 1/(x2 + 1)
Maximum: x = ∞, Minimum: x = −∞
Maximum: x = 0, Minimum: x = −∞
Maximum: x = 0, Minimum: x = N/A
Maximum: x = N/A, Minimum: x = N/A
Find the maximum and minimum for f on the given interval:
f (x) = 1/(x2 − 4) on [−1, 1]
Maximum: x = 0, Minimum: x = −1 and 1
Maximum: x = -0.25, Minimum: x = −0.333 and 0.333
Maximum: x = -0.25, Minimum: x = −0.333
Maximum: x = 0, Minimum: x = −1
Find the absolute maximum and minimum for f on the given interval.
f (x) = 1/(x2 − 4) on [−1, 4]
Maximum: x = 0, Minimum: x = −1 and 4
Maximum: x = −1, Minimum: x = 4
Maximum: x = 0, Minimum: x = −1
Maximum: N/A, Minimum: N/A
Select a polynomial function which has a local maximum at x = 1 and a local minimum at x = −1.
y = (x + 1)(x − 1)
y = (x + 1)2(x − 1)3
y = x3 − 3x + 3
y = x3 + 3x + 3
Suppose f is non-constant, continuous and differentiable on [a, b], and suppose f(a) = f(b). True or False, the MVT guarantees there is an absolute maximum or a minimum value for f on [a, b] at an interior point.
True
False
Select an example of a function f (x) and an interval [a, b] for which Rolle's Theorem is satisfied for more than one value of c in the interval [a, b]
lnx on [1, 5]
ex on [0, 1]
x2 on [−2, 2]
sin(x) on [−2π, 2π]
Select an example of a continuous function which is defined for all x, such that the derivative of the function is never zero, and it has a single unique global maximum value and no minimum value
lnx
ex
−|x|
|x|
Let f(x) = x2 − 2x − 2. Find a value of c which satisfies the conclusion of the Mean Value Theorem for f on the interval [1, 3].
c = 1
c = 2
c = −1
c = −2
