. Let f(x) = 3x^2 + 5x. Using the limit definition of derivative
prove that f '(x) = 6x + 5
Then, Find the tangent line of f(x) at x = 3
Finally, Find the average rate of change between x = −1 and x =
2
Use the ε-δ definition of limits to prove that lim x3 −2x2 −2x−3
= −6. 3 markx→1
Hint: This question needs students have a thorough understanding
of the proof by the ε-δdefinition as well as some good knowledge of
what is learnt in Math187/188 and in high school, such as long
division, factorization, inequality and algebraic
manipulations.
End of questions.
1. Use the ε-δ definition of continuity to prove that (a) f(x) =
x 2 is continuous at every x0. (b) f(x) = 1/x is continuous at
every x0 not equal to 0.
3. Let f(x) = ( x, x ∈ Q 0, x /∈ Q (a) Prove that f is
discontinuous at every x0 not equal to 0. (b) Is f continuous at x0
= 0 ? Give an answer and then prove it.
4. Let f and g...
Use the definition of absolute value and a proof by cases to
prove that for all real numbers x, | − x + 2| = |x − 2|. (Note:
Forget any previous intuitions you may have about absolute value;
only use the rigorous definition of absolute value to prove this
statement.)
Use the definition of the partial current to prove the linear
extrapolation distance for a vacuum boundary condition (the
distance where the flux function goes to zero) is d = 2/3λ , where
λ is the mean free math defined as λ = 1/Σtr = 3D (D is the
diffusion coefficient)
Prove that 1^3 + 2^3 + · · · + n^3 = (1 + 2 + · · · + n)^2 for
every n ∈ N. That is, the sum of the first n perfect cubes is the
square of the sum of the first n natural numbers. (As a student, I
found it very surprising that the sum of the first n perfect cubes
was always a perfect square at all.)
Use induction to prove
Let f(x) be a polynomial of degree n in Pn(R). Prove that for
any g(x)∈Pn(R) there exist scalars c0, c1, ...., cn such that
g(x)=c0f(x)+c1f′(x)+c2f′′(x)+⋯+cnf(n)(x), where f(n)(x)denotes the
nth derivative of f(x).