%3D 8. Use the Intermediate Value Theorem to prove that the equation e~² + x² = 4 has a real root.
Percentage
A percentage is a number indicated as a fraction of 100. It is a dimensionless number often expressed using the symbol %.
Algebraic Expressions
In mathematics, an algebraic expression consists of constant(s), variable(s), and mathematical operators. It is made up of terms.
Numbers
Numbers are some measures used for counting. They can be compared one with another to know its position in the number line and determine which one is greater or lesser than the other.
Subtraction
Before we begin to understand the subtraction of algebraic expressions, we need to list out a few things that form the basis of algebra.
Addition
Before we begin to understand the addition of algebraic expressions, we need to list out a few things that form the basis of algebra.
![7. Use derivatives to show that
a. (1+x) In(1 +x) > arctan x for x > 0.
b. arcsin V1 – x² + arccos x = T for x E (-1,0).
8. Use the Intermediate Value Theorem to prove that the equation e~ª + x² = 4 has a
real root.
x" – 1
for r > 0
9. Use Cauchy's form of the Mean Value Theorem to prove that Inx <
and r > 1.
12x + 5. Find the intervals on which f is
10. Let f : [-3, 2] → R, ƒ(x) = 2r + 3x2
increasing or decresing. Find its maximum and minimum, and prove that there is c E (1,2)
such that f'(c) = 11.
COS na
11. Show that the sequence of functions fn(x) =
n
converges uniformly on R.
12. Let f : [a, b] → [a, b] be continuous. Show that there is xo E [a, b] with f(xo) = xo.
13. Let f : (0, 7) → R, ƒ (x) = max{cos x, cos³ x}. Is f differentiable? Explain.
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