Zadanie 12a
First suppose is finite. (i) Show that b=2a−4. (ii) Find the limit in terms of a.
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First suppose is finite. (i) Show that b=2a−4. (ii) Find the limit in terms of a.
Now instead suppose f has a vertical asymptote, oblique asymptote y=x−5, and at least one point of zero gradient. State its vertical asymptote.
Now instead suppose f has a vertical asymptote, oblique asymptote y=x−5, and at least one point of zero gradient. Show that a=−3.
Now instead suppose f has a vertical asymptote, oblique asymptote y=x−5, and at least one point of zero gradient. Show that b ≥ −9.
For a=−3 and b=1: (i) sketch y = f(∣x∣), marking all asymptote equations, its vertical-axis intercept, and all local extrema; (ii) solve f(∣x∣) < f(x).
Solve Q(z)=0, giving the roots (i) in Cartesian form; (ii) in modulus–argument form.
Solve P(z)=0 in modulus–argument form.
Plot these six roots on an Argand diagram. Label them z₁,…,z₆ in counterclockwise order, starting with z₁ on the positive real axis.
Verify that z²−9 is a factor of P(z).
The already-specified quadratic factor is z2 − 9. P is the product of three quadratic polynomials with real coefficients. Find the other two quadratic factors.