CALCULUS Exercise
Ch1
1.1
7-10. Determine whether the curve is the graph of a function of x. If it is,
state the domain and range of the function.
1.2
3–4. Match each equation with its graph. Explain your choices. (Don't use
a computer or graphing calculator.)
1.3
39. Find 𝑓 ∘ 𝑔 ∘ ℎ.
𝑓(𝑥) = 3𝑥 − 2,
𝑔(𝑥) = sin 𝑥 ,
ℎ(𝑥) = 𝑥 2
1.5
5. For the function 𝑓 whose graph is given, state the value of each
quantity, if it exists. If it does not exist, explain why.
(a) lim 𝑓(𝑥)
(b) lim− 𝑓(𝑥)
(d) lim 𝑓(𝑥)
(e) 𝑓(3)
𝑥→1
(c) lim+ 𝑓(𝑥)
𝑥→3
𝑥→3
𝑥→3
1.6
20. Evaluate the limit, if it exists.
𝑡4 − 1
lim 3
𝑡→1 𝑡 − 1
47. The signum (or sign) function, denoted by 𝑠𝑔𝑛, is defined by
−1 if 𝑥 < 0
sgn 𝑥 = { 0 if 𝑥 = 0
1 if 𝑥 > 0
(a) Sketch the graph of this function.
(b) Find each of the following limits or explain why it does not exist.
(𝑖) lim+ sgn 𝑥
(𝑖𝑖) lim− sgn 𝑥
(𝑖𝑖𝑖) lim sgn 𝑥
(𝑖𝑣) lim | sgn 𝑥 |
𝑥→0
𝑥→0
𝑥→0
𝑥→0
1.7
14. Given that lim(5𝑥 − 7) = 3, illustrate Definition 2 by finding values
𝑥→2
of 𝛿 that correspond to 𝜀 = 0.1, 𝜀 = 0.05, and 𝜀 = 0.01.
1.8
3.(a) From the graph of 𝑓, state the numbers at which 𝑓 is discontinuous
and explain why.
3.(b) For each of the numbers stated in part (a), determine whether 𝑓 is
continuous from the right, or from the left, or neither.
42. Find the numbers at which 𝑓 is discontinuous. At which of these
numbers is 𝑓 continuous from the right, from the left, or neither?
Sketch the graph of 𝑓.
𝑥2 + 1
𝑓(𝑥) = { 3 − 𝑥
√𝑥
if 𝑥 ≤ 1
if 1 < 𝑥 ≤ 4
if 𝑥 > 4
CH2
2.1
9.(a) Find the slope of the tangent to the curve 𝑦 = 3 + 4𝑥 2 − 2𝑥 3 at
the point where 𝑥 = 𝑎.
(b) Find equations of the tangent lines at the points (1,5) and (2,3).
(c) Graph the curve and both tangents on a common screen.
31. Find 𝑓′(𝑎).
𝑓(𝑥) = 3𝑥 2 − 4𝑥 + 1
2.2
3. Match the graph of each function in (a) – (d) with the graph of its
derivative in I–IV. Give reasons for your choices.
2.3
17. Differentiate the function.
𝑦=
𝑥 2 + 4𝑥 + 3
√𝑥
44. Differentiate.
𝑓(𝑥) =
𝑎𝑥 + 𝑏
𝑐𝑥 + 𝑑
2.4
21. Find an equation of the tangent line to the curve at the given point.
𝑦 = sin 𝑥 + cos 𝑥 ,
(0,1)
48. Find the limit.
sin(𝑥 2 )
lim
𝑥→0
𝑥
2.5
5. Write the composite function in the form 𝑓(𝑔(𝑥)). [Identify the inner
function 𝑢 = 𝑔(𝑥) and the outer function 𝑦 = 𝑓(𝑢). Then find the
derivative 𝑑𝑦/𝑑𝑥.
𝑦 = √sin 𝑥
2.6
17. Find 𝑑𝑦/𝑑𝑥 by implicit differentiation.
√𝑥𝑦 = 1 + 𝑥 2 𝑦
39. If 𝑥𝑦 + 𝑦 3 = 1, find the value of 𝑦 ′′ at the point where 𝑥 = 0.
2.9
2. Find the linearization 𝐿(𝑥) of the function at 𝑎.
𝑓(𝑥) = sin 𝑥 ,
𝑎 = 𝜋/6
CH3
3.1
5-6. Use the graph to state the absolute and local maximum and minimum
values of the function.
29. Find the critical numbers of the function.
𝑓(𝑥) = 5𝑥 2 + 4𝑥
3.2
23. (a) Show that a polynomial of degree 3 has at most three real roots.
(b) Show that a polynomial of degree 𝑛 has at most 𝑛 real roots.
3.3
31. The graph of the derivative 𝑓 ′ of a continuous function 𝑓 is shown.
(a) On what intervals is 𝑓 increasing? Decreasing?
(b) At what values of 𝑥 does 𝑓 have a local maximum? Local
minimum?
(c) On what intervals is 𝑓 concave upward? Concave downward?
(d) State the 𝑥-coordinate(s) of the point(s) of inflection.
(e) Assuming that 𝑓(0) = 0, sketch a graph of 𝑓.
36. (a) Find the intervals of increase or decrease.
(b) Find the local maximum and minimum values.
(c) Find the intervals of concavity and the inflection points.
𝑔(𝑥) = 200 + 8𝑥 3 + 𝑥 4
3.4
25. Find the limit or show that it does not exist.
𝑥 4 − 3𝑥 2 + 𝑥
lim
𝑥→∞ 𝑥 3 − 𝑥 + 2
31. Find the limit or show that it does not exist.
lim 𝑥 sin
𝑥→∞
1
𝑥
3.5
1. Use the guidelines of this section to sketch the curve.
𝑦 = 𝑥3 + 𝑥
3.7
10. The rate (in mg carbon/m3 /h) at which photosynthesis takes place for
a species of phytoplankton is modeled by the function
100𝐼
𝑃= 2
𝐼 +𝐼+4
where 𝐼 is the light intensity (measured in thousands of foot-candles).
For what light intensity is 𝑃 a maximum?
3.8
7. Use Newton's method with the specified initial approximation 𝑥1 to
find 𝑥3 , the third approximation to the root of the given equation.
(Give your answer to four decimal places.)
2
− 𝑥 2 + 1 = 0,
𝑥
𝑥1 = 2
3.9
14. Find the most general antiderivative of the function. (Check your
answer by differentiation.)
5 − 4𝑥 3 + 2𝑥 6
𝑔(𝑥) =
𝑥6
30. Find 𝑓.
𝑓 ′ (𝑥) = 5𝑥 4 − 3𝑥 2 + 4,
𝑓(−1) = 2
CH4
4.1
25. Determine a region whose area is equal to the given limit. Do not
evaluate the limit.
𝑛
lim ∑
𝑛→∞
𝑖=1
𝜋
𝑖𝜋
tan
4𝑛
4𝑛
4.2
33. The graph of 𝑓 is shown. Evaluate each integral by interpreting it in
terms of areas.
2
(b) ∫0 𝑓 (𝑥) 𝑑𝑥
5
7
(d) ∫0 𝑓 (𝑥) 𝑑𝑥
(a) ∫0 𝑓 (𝑥) 𝑑𝑥
9
(c) ∫5 𝑓 (𝑥) 𝑑𝑥
4.3
15. Use Part 1 of the Fundamental Theorem of Calculus to find the
derivative of the function.
3𝑥+2
𝑦=∫
1
20. Evaluate the integral.
𝑡
𝑑𝑡
1 + 𝑡3
1
∫ 𝑥 100 𝑑𝑥
−1
4.4
25. Evaluate the integral.
𝜋
∫ (4 sin 𝜃 − 3 cos 𝜃) 𝑑𝜃
0
26. Evaluate the integral.
2
1
4
∫ ( 2 − 3 ) 𝑑𝑥
𝑥
1 𝑥
4.5
7. Evaluate the indefinite integral.
∫ 𝑥 √1 − 𝑥 2 𝑑𝑥
15. Evaluate the indefinite integral.
∫ cos(1 + 5𝑡) 𝑑𝑡
47. Evaluate the definite integral.
2
∫ 𝑥 √𝑥 − 1 𝑑𝑥
1