1. Use Riemann sums to compute the integral xdx. Hint. Divide the interval [0, 1] into n equal parts.

Calculus: Early Transcendentals
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Author:James Stewart
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Chapter1: Functions And Models
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9 of 27
1. Use Riemann sums to compute the integral xdx. Hint. Divide the
interval [0, 1] into n equal parts.
2. Find the integrals using the Fundamental Theorem of Calculus.
(»)
sin xdx.
(b)
cos xdx.
1
-dx.
cos? x
1
dx.
(d)
3. Show that
| Vi+x*dx < vV2.
Hint. Use the fact that if m < f(x) < M, then m(b – a) < S. f(x)dx < M(b – a).
4. Use the Cauchy-Schwarz inequality show
dt
b - a
Vab
5. Prove that the Mean Value Theorem for integrals (Theorem 1.3.6) is a
consequence of the Mean Value Theorem for derivatives applied to the
function F(x) = S" f(t)dt.
6. Let f' be continuous on [a, b). Prove that the Mean Value Theorem for
derivatives, is a consequence of the Mean Value Theorem for integrals
applied to f'.
7. Find the mean value of f (x)
= a.x + b in the interval [x1, x2).
1
8. Find the mean value of f(x) = x³ in the interval [0, 1].
9. Find the mean value of f(x) = V in the interval [0, 1].
%3D
10. Find the derivative F'(x), where
cx³
(a) F(x) =
cos tdt. (b) F(x) = |.
In tdt.
11. Let f(x) = S (1+ sin(sin t))đt. Find (f-1)'(0).
12. Show that
S sin t?dt
lim
1
3
13. Let f : [0, a] → R be a continuous function and
SE SE F(t)dt if x > 0,
g(x) =
f (0)
if x = 0.
Show that g is continuous at x = 0. Moreover, show that if f is differ-
entiable at x = 0, then g is differentiable on [0, a].
Transcribed Image Text:12:29 Done 9 of 27 1. Use Riemann sums to compute the integral xdx. Hint. Divide the interval [0, 1] into n equal parts. 2. Find the integrals using the Fundamental Theorem of Calculus. (») sin xdx. (b) cos xdx. 1 -dx. cos? x 1 dx. (d) 3. Show that | Vi+x*dx < vV2. Hint. Use the fact that if m < f(x) < M, then m(b – a) < S. f(x)dx < M(b – a). 4. Use the Cauchy-Schwarz inequality show dt b - a Vab 5. Prove that the Mean Value Theorem for integrals (Theorem 1.3.6) is a consequence of the Mean Value Theorem for derivatives applied to the function F(x) = S" f(t)dt. 6. Let f' be continuous on [a, b). Prove that the Mean Value Theorem for derivatives, is a consequence of the Mean Value Theorem for integrals applied to f'. 7. Find the mean value of f (x) = a.x + b in the interval [x1, x2). 1 8. Find the mean value of f(x) = x³ in the interval [0, 1]. 9. Find the mean value of f(x) = V in the interval [0, 1]. %3D 10. Find the derivative F'(x), where cx³ (a) F(x) = cos tdt. (b) F(x) = |. In tdt. 11. Let f(x) = S (1+ sin(sin t))đt. Find (f-1)'(0). 12. Show that S sin t?dt lim 1 3 13. Let f : [0, a] → R be a continuous function and SE SE F(t)dt if x > 0, g(x) = f (0) if x = 0. Show that g is continuous at x = 0. Moreover, show that if f is differ- entiable at x = 0, then g is differentiable on [0, a].
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