Fun Worksheet #1
1. A function f is continuous on the closed interval [-3, 3] such that f(-3) = 4 and f(3) = 1. The functions f’ and f’’ have
the following properties below.
a.
b.
c.
d.
2. Let
a.
b.
c.
x
-3 < x < -1
x = -1
-1 < x < 1
x=1
f’(x)
+
DNE
0
f’’(x)
+
DNE
+
0
What are the x-coordinates of the absolute minimum of f? Justify your answer.
What are the x-coordinates of the absolute maximum of f? Justify your answer.
What are the x-coordinates of all points of inflection of f? Justify your answer.
Sketch a graph of f that satisfies the given properties of f.
1<x<3
-
f’’(x) = 1 + (x)sinx + (lnx)cosx on the interval 1 < x < 7.
On what intervals is the graph concave up? Justify.
Find the x-coordinates of all relative minimums for function f’(x). Justify.
Find the x-coordinates of all points of inflection on the graph of f’(x). Justify.
3
2
3. A particle moves along the x-axis so that at time t its position is given by 𝑥(𝑡) = 𝑡 − 6𝑡 + 9𝑡 + 11.
a. What is the particle’s velocity at t = 0?
b. During what time intervals is the particle moving left? Justify.
c. When is the particle’s speed slowing down? Justify.
d. What is the total distance traveled by the particle from t = 0 to t = 2?
3
2
4. Find the values of a, b, and c such that the function 𝑓(𝑥) = 𝑥 + 𝑎𝑥 + 𝑏𝑥 + 𝑐 has a relative extrema at (1, 5) and a
point of inflection at (2, 3).
3
2
5. For what value(s) of x does the slope of the tangent line to the curve 𝑦 =− 𝑥 + 3𝑥 + 1 take on its largest value?
Justify.
6. Let f be a three times differentiable function and suppose for some number c that f’(c) = f’’(c) = 0 but f’’’(c) > 0. Does
f have a relative minimum, relative maximum, or a point of inflection at x = c? Explain your reasoning.
7. A particle moves along the x-axis so that its velocity at time t is given by 𝑣(𝑡) =− (𝑡 + 1)𝑠𝑖𝑛(
2
𝑡
2
). At t = 0, the
particle is at position x = 1.
a. Find the acceleration of the particle at time t = 2.
b. Is the speed of the particle increasing at t = 2? Why or why not?
c. Find all times t in the open interval 0 < t < 3 when the particle changes direction. Justify your answer.
8. The accompanying figures show the graph of the derivative of a function f continuous on its domain, which is the set
of real numbers. From each graph determine the following.
i. the critical numbers of f
ii. the intervals on which f is increasing
iii. the intervals on which f is decreasing
iv. the relative extrema of f
v. the point(s) of inflection of f