Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter14: Discrete Dynamical Systems
Section14.3: Determining Stability
Problem 13E: Repeat the instruction of Exercise 11 for the function. f(x)=x3+x For part d, use i. a1=0.1 ii...
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![14. Show there does not exist a continuously differentiable function f on [0, 21 such that
f(0) = -1, f(2) = 4, and f (x) < 2 for 0<x < 2. (Apply the Fundamental Theorem.)
15. Iff: R R is continuous and e> 0, define g : R R by g(x) := ()dt. Show that g is
differentiable on R and find g'(x).
16. If f: [0, 1]→R is continuous and f = f for all x E [0, 1], show that f(x) =0 for all
xE (0, 1].
17. Use the following argument to prove the Substitution The orem 7.3.8. Define F(u) = [gf(x)dx
for u E I, and H(t) := F((t)) for tE J. Show that H'(1) =f(p(t))(t) for te J and that
f(x)dx = F(4(F)) = H(B) =
Ja
18. Use the Substitution Theorem 7.3.8 to evaluate the following integrals.
(a) VI+fdr,
TP(1+)dt = 4/3,
(b)
cos i](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd4f4edce-3df2-42bf-9fa8-58dcd89039a8%2F03efb716-9858-400a-9d40-a47f07aed5bc%2Fl0gh5j_processed.jpeg&w=3840&q=75)
Transcribed Image Text:14. Show there does not exist a continuously differentiable function f on [0, 21 such that
f(0) = -1, f(2) = 4, and f (x) < 2 for 0<x < 2. (Apply the Fundamental Theorem.)
15. Iff: R R is continuous and e> 0, define g : R R by g(x) := ()dt. Show that g is
differentiable on R and find g'(x).
16. If f: [0, 1]→R is continuous and f = f for all x E [0, 1], show that f(x) =0 for all
xE (0, 1].
17. Use the following argument to prove the Substitution The orem 7.3.8. Define F(u) = [gf(x)dx
for u E I, and H(t) := F((t)) for tE J. Show that H'(1) =f(p(t))(t) for te J and that
f(x)dx = F(4(F)) = H(B) =
Ja
18. Use the Substitution Theorem 7.3.8 to evaluate the following integrals.
(a) VI+fdr,
TP(1+)dt = 4/3,
(b)
cos i
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