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Review Sheet for Second Midterm
Mathematics 1300, Calculus 1
1. Suppose f (2) = 3 and f 0 (2) = 1. Find f (−2) and f 0 (−2) if f is assumed to be even.
2. Given all of the following information about a function f , sketch its graph.
• f (x) = 0 at x = −5, x = 0, and x = 5
• lim f (x) = ∞
x→−∞
• lim f (x) = −3
x→∞
0
• f (x) = 0 at x = −3, x = 2.5, and x = 7
3. Given the graph of y = f (x) shown, sketch the graph of the derivative.
0
−2
−1
0
1
2
3
4
5
x
4. Given the following table of values, estimate f 0 (0.6) and f 0 (0.5); then use these to
estimate f 00 (0.6).
x
f (x)
0 0.2 0.4 0.6 0.8 1.0
3.7 3.5 3.5 3.9 4.0 3.9
5. If possible, draw the graph of a differentiable function with domain [0, 6] satisfying
f 0 (x) > 0 for x < 1, f 0 (x) < 0 for x > 1, f 0 (x) > 0 for x > 4, f 00 (x) > 0 for x < 3, and
f 00 (x) < 0 for x > 3.
6. If y = f (x) is the graph shown below, sketch the graphs of y = f 0 (x) and y = f 00 (x).
Estimate slopes as needed.
1
6
5
4
3
2
1
0
−1
−2
−4 −3 −2 −1 0
1
2
3
4
5
x
7. The graph of y = f 0 (x) is shown. Where is f (x) greatest? Least? Where is f 0 (x)
greatest? Least? Where is f 00 (x) greatest? Least? Where is f (x) concave up? Concave
down?
0
−2
−1
0
x
1
2
8. Name three different reasons a function can fail to be differentiable at a point.
9. For what values of x is the graph of y = x5 − 5x both increasing and concave up?
10. Where does the tangent line to y = 2x through (0, 1) intersect the x-axis?
11. If g(x) = ex f (x), find and simplify g 00 (x).
12. If f (x) = 4x3 + 6x2 − 23x + 7, find the intervals on which f 0 (x) ≥ 1.
13. Let f (x) and g(x) be the functions graphed below.
2
4
4
3
3
2
2
1
1
0
0
−1
−1
−2
−2
−3
−3
−4
−4 −3 −2 −1
0
x
1
2
3
−4
−4 −3 −2 −1
4
0
x
1
2
3
4
Let h(x) = f (x)g(x), let j(x) = x2 f (x), let k(x) = f (x2 ), let p(x) = f (x)/g(x), let
q(x) = f (g(x)), and r(x) = g(g(x)).
(a) Estimate h0 (1), h0 (0), p0 (0), q 0 (0), r0 (1), and r0 (2).
(b) Estimate j 0 (−1) and k 0 (−1).
(c) Estimate all values of x for which y = r(x) has a horizontal tangent line.
14. Using the information in the table, find:
(a) h(4) if h(x) = f (g(x))
(b) h0 (4) if h(x) = f (g(x))
x
f (x)
f 0 (x)
g(x)
g 0 (x)
(c) h(4) if h(x) = g(f (x))
(d) h0 (4) if h(x) = g(f (x))
(e) h0 (4) if h(x) = g(x)/f (x)
1
3
1
2
4
2
2
4
1
2
3
1
2
4
3
4
4
3
3
1
(f) h0 (4) if h(x) = f (x)g(x)
15. A graph of f (x) is shown below. It is piecewise linear. The table below gives values of
g(x) and g 0 (x).
4
x
0
g(x) 2
g 0 (x) 3
f (x)
1 2 3 4
a) Given h(x) = f (x)g(x), find h0 (1).
3
1
5
4
2
9
2
3 4
11 8
-3 -4
f (x)
, find k 0 (3).
g(x)
g(x)
c) Given `(x) = √ , find `0 (4).
x
d) Given m(x) = g(f (x)), find m0 (3).
b) Given k(x) =
16. On what interval(s) is the function
f (x) =
(5x + 2)3
(2x + 3)5
increasing?
17. If g(2) = 3 and g 0 (2) = −4, find f 0 (2) for the following:
(a) f (x) = x2 − 4g(x)
(b) f (x) =
x
g(x)
(c) f (x) = x2 g(x)
(d) f (x) = g(x)2
(e) f (x) = x sin (g(x))
(f) f (x) = x2 ln (g(x))
18. On what interval(s) is the function
f (x) = (x + 3)e2x
decreasing? On what intervals is it concave down?
19. The depth D (in inches) of water in a swimming pool being filled is given by D = f (t)
where t is in minutes since the hose was turned on.
(a) Is f 0 (t) positive or negative? What are the units of f 0 (t)?
(b) What would it mean to say that f (20) = 5 and f 0 (20) = 0.1?
(c) What are the units of the number 0.1 above?
20. If the graph of y = g(x) is shown below, sketch the graph of y = g 0 (x).
21. A study of 100 Boulder residents concluded that the number N of “happy” residents
on any given day was given by N = f (h), a function of the number of hours of sunshine
(h) during that day.
4
(a) Suppose we found that f (3) = 40. Explain the meaning in a sentence, using
units.
(b) Suppose that f 0 (8) = 1. What are the units of 8? What are the units of 1?
Explain the meaning in a sentence.
22. Consider the function
√
x
f (x) = √
−x
when x ≥ 0
when x < 0
At what x-values, if any, is this function not continuous? At what x-values, if any, is
this function not differentiable, and why?
23. If the position of a particle at time t is given by the formula s(t) = t3 − t, what is the
velocity of the particle at time t = 1?
24. A rock falling from the top of a vertical cliff drops a distance of s(t) = 16t2 feet in t
seconds. What is its speed at time t? What is its speed when it has fallen 64 feet?
25. If f (x) = x2 + 1 and g(x) = 5 − x, find:
(a) h0 (1) if h(x) = f (x) · g(x)
(b) j 0 (2) if j(x) =
f (x)
g(x)
(c) k 0 (3) if k(x) = f (g(x))
26. Find constants a and b so that the following piecewise function is differentiable.
(
ax + b x < 2,
f (x) =
.
x3 + 10 x ≥ 2.
27. A boat at anchor is bobbing up and down in the sea. The vertical distance, y, in feet,
between the sea floor and the boat is given as a function of time, t, in minutes by
y = 15 + 6 sin(2πt)
dy
.
dt
dy
(b) Find
when t = 65 . Explain in an English sentence what this means in terms of
dt
the movement of the boat. Include units.
(a) Find
28. Find
dy
dx
if
x3 + y 3 − 4x2 y = 0.
29. Find the slope of the curve y + sin y + x2 = 9 at (3, 0).
30. Find the equations for the lines tangent to the graph of xy + y 2 = 4 when x = 3.
31. Consider the curve x2 + 2xy + 5y 2 = 4. At what point(s) is the tangent line to this
curve horizontal? At what point(s) is the tangent line to this curve vertical? At what
points is the slope of the tangent line equal to 2?
5
Derivative Practice.
Find f 0 (x).
1. f (x) =
x2 + 1
5
20. f (x) = (x2 + 1)1000
21. f (x) = ln(cos x)
2. f (x) = π 3
3. f (x) =
√
x+
22. f (x) = (4x2 + 1)5
1
3x
2
3
4
5
4. f (x) =
1+x +x +x +x +x
x3
5. f (x) =
1
x2
23. f (x) = arctan ln cos x2
6
24. f (x) = ex
2 +x+1
25. f (x) = sin (cos (tan x))
√
26. f (x) = x3 − 2x + 5
6. f (x) = (x + 1)(2x − 1)
27. f (x) =
7. f (x) = xex
8. f (x) = sin x cos x
1
(x5 − x + 1)9
(2x + 3)3
(4x2 − 1)8
29. f (x) = x5 cos x1
28. f (x) =
9. f (x) = (ln x)ex
10. f (x) = kxn
30. f (x) = arcsin (sin x)
2x − 1
11. f (x) =
x+3
12. f (x) = (2x7 − x2 ) ·
31. f (x) = arctan (4x + 3)
x−1
x+1
32. f (x) = (arcsin x)(arctan x)
33. f (x) = arcsin(x tan x)
x2 + 1
13. f (x) =
x+1
14. f (x) =
34. f (x) = earcsin(4x
sec(x)
1+tan x
35. f (x) = ln (arcsin x) + xex
15. f (x) = 2 sin2 (x)
36. f (x) = tan2 (arcsin 1)
16. f (x) = cos2 (x) + sin2 (x)
37. f (x) =
17. f (x) =
sin x cos x
1 + x tan x
1+x
1−x
38. f (x) = sec ln x
q
p
√
39. f (x) = x + x + x
18. f (x) = x2 cos x + 4 sin x
19. f (x) =
2)
ex
ln x
40. f (x) = xln x
6
2