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Apostol, Calculus, Volume 1 solutions.

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I - Introduction
I.1.4 – Exercises using the method of exhaustion to calculate areas under various curves.
#1, #2, #3
I.2.5 – Exercises proving basic properties of sets
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10, #11, #12, #13, #14, #15, #16, #17, #18, #19, #20
I.3.3 – Exercises proving some consequences of the field axioms.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10
I.3.5 – Exercises proving some consequences of the order axioms.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10
I.3.12 – Exercises proving various properties of the Real number system.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10, #11, #12
I.4.4 – Exercises on proofs by induction.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10, #11, #12
I.4.7 – Exercises on summation notation.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10, #11, #12, #13
I.4.9 – Exercises proving some properties of absolute values.
#1, #2, #3, #4
I.4.10 – More exercises on proofs by induction.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10, #11, #12, #13, #14, 15, #16, #17, #18, #19, #20, #21, #22, #23

1 - The Concepts of Integral Calculus
1.5 – Exercises on functions.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10, #11, #12
1.7 – Exercises on the axiomatic definition of area.
#1, #2, #3, #4, #5, #6
1.11 – Exercises on partitions and step functions.
#1, #2, #3, #4, #5, #6, #7, #8
1.15 – Exercises on integrals of step functions.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10, #11, #12, #13, #14, #15, #16, #17
1.26 – Exercises on definite integrals of polynomials.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10, #11, #12, #13, #14, #15, #16, #17, #18, #19, #20, #21, #22, #23, #24, #25, #26, #27, #28

2 - Some Applications of Integration
2.4 – Exercises on the area of graphs.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10, #11, #12, #13, #14, #15, #16, #17, #18, #19, #20, #21
2.8 – Exercises on integrals of trigonometric functions.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10, #11, #12, #13, #14, #15, #16, #17, #18, #19, #20, #21, #22, #23, #24, #25, #26, #27, #28, #29, #30, #31, #32, #33, #34
2.11 – Exercises on polar coordinates.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10, #11, #12, #13, #14, #15
2.13 – Exercises on using integration techniques to calculate volume.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10, #11, #12, #13, #14, #15, #16, #17, #18
2.15 – Exercises on applications of integration to calculate work.
#1, #2, #3, #4, #5, #6, #7, #8, #9
2.17 – Exercises on computing the average value of a function.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10, #11, #12, #13, #14, #15, #16, #17, #18, #19, #20, #21, #22, #23, #24
2.19 – Exercises on integrals expressed as functions of the upper limit.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10, #11, #12, #13, #14, #15, #16, #17, #18, #19, #20, #21

3 - Continuous Functions
3.6 – Exercises on evaluating limits of basic functions.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10, #11, #12, #13, #14, #15, #16, #17, #18, #19, #20, #21, #22, #23, #24, #25, #26, #27, #28, #29, #30, #31, #32, #33
3.8 – Exercises on compositions of functions and limits of compositions of functions.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10, #11, #12, #13, #14, #15, #16, #17, #18, #19, #20, #21, #22, #23
3.11 – Exercises on the intermediate value theorem for continuous functions.
#1, #2, #3, #4, #5, #6
3.15 – Exercises on properties of inverses of functions.
#1, #2, #3, #4, #5, #6, #7, #8
3.20 – Exercises on the mean value theorem for integrals of continuous functions.
#1, #2, #3, #4, #5, #6, #7, #8

4 - Differential Calculus
4.6 – Exercises on derivatives of basic functions.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10, #11, #12, #13, #14, #15, #16, #17, #18, #19, #20, #21, #22, #23, #24, #25, #26, #27, #28, #29, #30, #31, #32, #33, #34, #35, #36, #37, #38, #39
4.9 – Exercises on geometric interpretation of the derivative of functions.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10, #11, #12, #13, #14, #15, #16
4.12 – Exercises on determining derivatives and on related rates and implicit differentiation.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10, #11, #12, #13, #14, #15, #16, #17, #18, #19, #20, #21, #22, #23, #24, #25, #26, #27, #28, #29, #30, #31, #32, #33, #34
4.15 – Exercises on extreme values and the mean value theorem for derivatives.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10
4.19 – Exercises on using derivatives to determine properties of the graph of a function.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10, #11, #12, #13, #14
4.21 – Exercises on application of differentiation to extremum problems.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10, #11, #12, #13, #14, #15, #16, #17, #18, #19, #20, #21, #22, #23, #24, #25, #26, #27, #28
4.23 – Exercises on partial derivatives.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10

5 - The Relation Between Integration and Differentiation
5.5 – Exercises on the first and second fundamental theorems of calculus.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10, #11, #12, #13, #14, #15, #16, #17, #18, #19, #20, #21, #22, #23, #24, #25, #26, #27, #28
5.8 – Exercises on evaluating integrals.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10, #11, #12, #13, #14, #15, #16, #17, #18, #19, #20, #21, #22, #23, #24, #25, #26, #27
5.10 – Exercises on integration by parts.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10, #11, #12, #13, #14, #15, #16, #17, #18, #19, #20, #21
5.11 – Exercises reviewing integration.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10, #11, #12, #13, #14, #15, #16, #17, #18, #19, #20, #21, #22, #23, #24, #25, #26, #27, #28, #29, #30, #31, #32, #33, #34, #35, #36

6 - The Logarithm, the Exponential, and the Inverse Trigonometric Functions
6.9 – Exercises on differentiation and integration of logarithm.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10, #11, #12, #13, #14, #15, #16, #17, #18, #19, #20, #21, #22, #23, #24, #25, #26, #27, #28, #29, #30, #31, #32, #33, #34, #35, #36
6.11 – Exercises on polynomial approximation of logarithms.
#1, #2, #3, #4, #5
6.17 – Exercises on derivatives of exponential functions.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10, #11, #12, #13, #14, #15, #16, #17, #18, #19, #20, #21, #22, #23, #24, #25, #26, #27, #28, #29, #30, #31, #32, #33, #34, #35, #36, #37, #38, #39, #40, #41, #42, #43
6.19 – Exercises on the hyperbolic functions.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10, #11, #12, #13, #14, #15, #16, #17, #18, #19, #20, #21, #22, #23, #24, #25, #26
6.22 – Exercises on derivatives of inverse trigonometric functions.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10, #11, #12, #13, #14, #15, #16, #17, #18, #19, #20, #21, #22, #23, #24, #25, #26, #27, #28, #29, #30, #31, #32, #33, #34, #35, #36, #37, #38, #39, #40, #41, #42, #43, #44, #45, #46, #47
6.25 – Exercises on integration by partial fractions.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10, #11, #12, #13, #14, #15, #16, #17, #18, #19, #20, #21, #22, #23, #24, #25, #26, #27, #28, #29, #30, #31, #32, #33, #34, #35, #36, #37, #38, #39, #40
6.26 – Exercises reviewing integration of exponentials, logarithms, and inverse trigonometric functions.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10, #11, #12, #13, #14, #15, #16, #17, #18, #19, #20, #21, #22, #23, #24, #25, #26, #27, #28, #29, #30

7 - Polynomial Approximations to Functions
7.4 – Exercises on calculus of Taylor polynomials
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10
7.8 – Exercises on Taylor’s formula with remainder.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10
7.11 – Exercises on little o-notation and applications to indeterminate forms.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10, #11, #12, #13, #14, #15, #16, #17, #18, #19, #20, #21, #22, #23, #24, #25, #26, #27, #28, #29, #30, #31, #32, #33
7.13 – Exercises on L’Hopital’s rule.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10, #11, #12, #13, #14, #15, #16, #17, #18
7.17 – Exercises on evaluating limits involving exponentials and logarithms.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10, #11, #12, #13, #14, #15, #16, #17, #18, #19, #20, #21, #22, #23, #24, #25, #26, #27, #28, #29, #30, #31, #32

8 - Introduction to Differential Equations
8.5 – Exercises on first-order linear differential equations.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10, #11, #12, #13, #14, #15, #16, #17, #18, #19, #20
8.7 – Exercises on applying first-order linear differential equations to physical problems.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10, #11, #12, #13, #14, #15, #16, #17, #18
8.14 – Exercises on second-order linear equations with constant coefficients.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10, #11, #12, #13, #14, #15, #16, #17, #18, #19, #20, #21, #22, #23
8.17 – Exercises on non-homogeneous second-order linear equations with constant coefficients.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10, #11, #12, #13, #14, #15, #16, #17, #18, #19, #20, #21, #22, #23, #24, #25
8.19 – Exercises on physical problems leading to second-order linear differential equations.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10, #11, #12
8.22 – Exercises on integral curves.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10, #11, #12, #13, #14, #15
8.24 – Exercises on first-order separable differential equations.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10, #11, #12, #13, #14, #15, #16, #17, #18, #19, #20
8.26 – Exercises on similarity transforms of first-order differential equations.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10, #11
8.28 – Exercises reviewing differential equations.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10, #11, #12, #13, #14, #15, #16, #17, #18, #19, #20, #21, #22, #23, #24, #25, #26, #27, #28, #29, #30, #31

9 - Complex Numbers
9.6 – Exercises on complex numbers.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10, #11, #12
9.10 – Exercises on complex functions.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10, #11, #12, #13, #14, #15

10 - Sequences, Infinite Series, Improper Integrals
10.4 – Exercises on sequences.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10, #11, #12, #13, #14, #15, #16, #17, #18, #19, #20, #21, #22, #23, #24, #25, #26, #27, #28, #29, #30, #31, #32, #33, #34, #35
10.9 – Exercises on series.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10, #11, #12, #13, #14, #15, #16, #17, #18, #19, #20, #21, #22, #23, #24, #25
10.10 – Exercises on decimal expansions.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10
10.14 – Exercises on comparison test, limit comparison test, and integral test for convergence.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10, #11, #12, #13, #14, #15, #16, #17, #18, #19
10.16 – Exercises on root test and ratio test for convergence.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10, #11, #12, #13, #14, #15, #16, #17, #18
10.20 – Exercises on alternating series, and convergence tests of Dirichlet and Abel.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10, #11, #12, #13, #14, #15, #16, #17, #18, #19, #20, #21, #22, #23, #24, #25, #26, #27, #28, #29, #30, #31, #32, #33, #34, #35, #36, #37, #38, #39, #40, #41, #42, #43, #44, #45, #46, #47, #48, #49, #50, #51, #52
10.22 – Exercises reviewing sequences and series.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10, #11, #12, #13, #14, #15, #16, #17, #18
10.24 – Exercises on improper integrals.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10, #11, #12, #13, #14, #15, #16, #17, #18, #19, #20, #21, #22, #23, #24, #25

11 - Sequences and Series of Functions
11.7 – Exercises on power series and radius of convergence.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10, #11, #12, #13, #14, #15, #16, #17, #18, #19, #20
11.13 – Exercises on functions represented by power series.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10, #11, #12, #13, #14, #15, #16, #17, #18, #19, #20, #21, #22, #23, #24
11.16 – Exercises on power series and differential equations.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10, #11, #12, #13, #14, #15, #16, #17, #18, #19, #20, #21, #22

12 - Vector Algebra
12.4 – Exercises on vector algebra.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10, #11, #12
12.8 – Exercises on dot products and norms of vectors.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10, #11, #12, #13, #14, #15, #16, #17, #18, #19, #20, #21, #22, #23, #24, #25
12.11 – Exercises on projections and unit coordinate vectors.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10, #11, #12, #13, #14, #15, #16, #17, #18, #19, #20
12.15 – Exercises on spans, linear independence, and bases.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10, #11, #12, #13, #14, #15, #16, #17, #18, #19, #20
12.17 – Exercises on the vector space of n-tuples of complex numbers.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10

13 - Applications of Vector Algebra to Analytic Geometry
13.5 – Exercises on lines.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10, #11, #12
13.8 – Exercises on planes.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10, #11, #12, #13, #14
13.11 – Exercises on the cross product.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10, #11, #12, #13, #14, #15
13.14 – Exercises on the scalar triple product and Cramer’s rule.
(Note: In my edition of the text (which I think is the standard one) the numbering skips #8. I left a blank page under #8 so the problem numbers will still match up with the text.)
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10, #11, #12, #13, #14, #15, #16, #17, #18, #19, #20
13.17 – Exercises on normal vectors and Cartesian equations of planes.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10, #11, #12, #13, #14, #15, #16, #17, #18, #19, #20, #21, #22, #23, #24
13.21 – Exercises on conic sections.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10, #11, #12, #13
13.24 – More exercises on conic sections.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10, #11, #12, #13, #14, #15, #16, #17, #18, #19, #20, #21, #22, #23, #24, #25, #26, #27, #28, #29, #30, #31, #32, #33, #34, #35, #36, #37, #38
13.25 – Exercises reviewing conic sections.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10, #11, #12, #13, #14, #15, #16, #17, #18, #19, #20, #21, #22, #23, #24

14 - Calculus of Vector-Valued Functions
14.4 – Exercises on limits, derivatives, and integrals of vector-valued functions.
#1, #2, #3, #4, #5, #6, #7, #8, #9, #10, #11, #12, #13, #14, #15, #16, #17, #18, #19, #20,
14.7 – Exercises on tangents to curves and curvilinear motion.

14.9 – Exercises on the unit tangent and the principal normal.

14.13 – Exercises on arc length.

14.15 – Exercises on curvature

14.19 – Exercises on polar coordinates and cylindrical coordinates.

14.21 – Exercises reviewing vector-valued functions.

36 comments

  1. tom says:

    Rori – this blog and solutions has been very helpful in my math journey. The presentation is so clear and concise I think you could continue these endeavors for more books and ask for donations- but alas you probably have higher aspirations than this. I am almost done with Calc. 1 after more than 3 years- without answers I can’t imagine how much extra time it would take. I assumed that Apostol, Spivak and Courant covered the same material, and although they play well off each other, the combined span of the 3 is much greater than any one. Hopefully can read most of the books in your list, although a rough estimate of the time frame would be 20 years. Oh well, it’s the journey that counts. Thanks again!

  2. Jon Poler says:

    Hi Rori. This is amazing! Thank you so much for your effort!

    Suppose that someone wanted to work through Apostol, but doesn’t have time to complete every exercise exhaustively. Would you be able to comment on good heuristics for choosing a subset of the problems to solve while still getting 80-90% of the understanding? I’d like to work through vol 1 and 2.

    • Mal says:

      Recommend Open Courseware at MIT – Calculus with Theory and Multivariable Calculus with Theory. They have readings / recitations/ assignments and exams based on these texts.

  3. Manvi says:

    Thanks for the solutions, it was really helpful. I hope you can also help me with chapter 16 solutions. Thanks!

  4. Otay says:

    On the partial fractions exercises in 6.25 that involve trigonometry, how do you know when to use the u=tan(x/2) substitution? Or more importantly, when not to? (I would have never thought of the manipulation you used on #31. I tried u=tan(x/2) and failed.)

    P.S. Thank you so very much for all of this. Bless you.

  5. Gabriel Rangel says:

    Apostol’s book never ceases to make me feel stupid, which is why I chose it. But I don’t think I could get through without your solutions, thank you.
    I hope you get to vol 2 by the next semester.
    Keep up the good work!

  6. Stupendous says:

    I see that there are no more solutions from chapter 8 – Introduction to Differential Equations onwards. Also, are you going to make solutions for volume 2 as well? Thanks for creating the wonderful solution guide by the way!

    • RoRi says:

      Hi. I’m just finishing Chapter 7 tonight (still 2 more problems). I’ll start on Chapter 8 tomorrow. The plan is to average 5 problems per day over the course of each month (if you look at the archives on the side you’ll see that each month has 150 or 155 entries), which means I’ll finish volume 1 on June 23. I will definitely do volume 2, but maybe not right away. I was thinking of blogging the solutions to either a linear algebra book or a topology book after I finish volume 1. (It makes more sense for my coursework to do one of those rather than a multivariable calculus book.)

      No problem on creating the solutions. Please let me know if you find any mistakes or typos or anything.

      • tom says:

        June 23 is the beginning of summer- well timed! I also am looking forward to being finished by about beginning of summer, with the trifecta of calculus books (spivak, courant and apostol; admittingly dependent on your work). I can’t tell you how valuable this stuff is to a self learner. I have been thru spivak twice now due to the sheer volume of questions, in spite of having solutions; the first time thru I was little more then a proof confirming robot while the second time I am able to grasp the landscape, so to speak. My next books will probably be analysis by Pugh, and Taos. Any interest in working those books?

      • RoRi says:

        I might do an analysis book at some point, but it will be a while. Currently, the next three books I plan to do are Hoffman & Kunze’s Linear Algebra, Willard’s General Topology, and Volume 2 of Apostol. (Not necessarily in that order.) If I can maintain a 5 problem per day pace, doing all of the exercises in those three will take me until December 8, 2017 (modulo possible counting errors). So… even if I do an analysis book it’s going to be a long time before I get started. There are a number of solution sets (of varying quality) to Rudin’s Principles of Mathematical Analysis floating around online. I think there are also solutions out there for Apostol’s Mathematical Analysis.

        Also, I think you asked a question about this exercise back in November? I did finally get around to updating that. I don’t have anything insightful to say about the motivation for the problem, but at least the solution should now be mathematically correct (I hope).

      • tom says:

        I will be looking forward to seeing your next work. I am geared to attack analysis soon which I give myself until Dec. 2017- conveniently the time you stated for the 3 books. Certainly the Apostol II and the Linear Algebra look very enticing. Perhaps one day the topo….

  7. RC Fernandez says:

    This is gold. The only material i could find on the internet.

    I would love to print on a paper, do you have any compiled version ? Such as pdf, txt or whatever. It will give me a hell of a job to put it all together in one file just to print. Pls say that you have.

    • RoRi says:

      I’m really sorry, but I do not have a PDF of the blog. All of the pages are generated by WordPress and the images with math are inserted using a plug-in. I should probably make a FAQ that explains this at some point since people are asking for PDFs. I also want to keep it in blog format so that people can leave comments and follow the links to previous results when they are necessary.

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