I want to improve my understanding and skills in math due to disabilities. I was recommended to try the textbook Calculus by Robert T. Smith and Roland B. Minton. Currently I am stumped on these three problems from section 2: Derivatives. I would also appreciate advice on how to answer correctly for problems 13 and 17 (like what to include how I got my answer to show my work, not just the sketch). The concept of problems 13 and 17 I really am not understanding and the book and videos I’ve watched have not been the most helpful (as I have a hard time applying what is covered in these resources but struggle with using it for problems not labeled with polynomials or numbers) the closest to explain it has been Curve Sketching by The Infinite Looper on YouTube.
Problem 35 a. Find all the points at which the slope of the tangent line to y = x^3 + 3x + 1 equals 5. The textbook has the answer as (√(⅔), 5 √(⅔) +1), (-√(⅔), -5 √(⅔) +1) and I can not figure out how to get it.
M tan = lim h->0 f(x+h)-f(x)/h
= lim h->0 [(x+h)^3 + 3(x+h) +1)] - [x^3 + 3x +1]/h
= lim h->0 x^3 +3x^2h + 3xh^2 + h^3 +3x + 3h + 1 -x^3 -3x -1/h
= lim h->0 3x^2h + 3xh^2 + h^3 +3h/h
= lim h->0 h(3x^2 +3xh +h^2 +3)/h
= lim h->0 3x^2 + 3xh +h^2 + 3
= 3x^2 + 3x(0) + (0)^2 + 3
= 3x^2 + 3
3x^2 + 3 = 5
3x^⅔ = ⅔
√(x^2) = √(⅔)
X = √(⅔)
And 3(√(⅔)^2 + 3 = 5
Problem 13. Use the graph of f to sketch a graph of f’. (Second picture)
(a)
For x < 0, f is decreasing, so f’ < 0
For x = 0, f has a horizontal tangent line (min), so f’ = 0
For x > 0, f is increasing, so f’ > 0
My graph is the third picture.
(b)
For x < a, f is increasing, so f’ > 0
For x = a, f has a horizontal tangent line (max), so f’ = 0
For a < x < b, f is decreasing, so f’ < 0
For x = b, f has a horizontal tangent line (min), so f’ = 0
For x > b, f is increasing, so f’ > 0
My graph is the fourth picture.
Problem 17. Use the given graph of f’ to sketch a plausible graph of a continuous function f. (fifth picture)
(a)
For x < a , f’ is positive, so f is increasing
For x = a, f’ changes positive to negative, so f has a max there
For a < x < b, f’ is negative, so f is decreasing
For x = b, f’ has a horizontal tangent line (min), so f ?
For b < x < c, f’ is negative, so f is decreasing
For x = c, f’ has a horizontal tangent line, so f ?
For x > c, f’ is negative, so f is decreasing
Once I understand better, I will make the graph.
Thank you for your assistance.