- Prove that if (n − 1)! + 1 is divisible by n (> 1), then n must be prime.

From the differential equation y′ = x + sin(y):a) A solution curve passes through the point (1, π/2). What is its slope at that point?b) Justify why for x > 1 the solutions are increasing.c) Show that the concavity of each solution has the function 1 + x cos(y) + 1/2 sin(2y).Justify each of the steps.d) A solution curve passes through the point (0, 0). Show that the curve has a minimumrelative at (0, 0).

Q/ Qfind the incidence matrix for the graph K₁ UC

What will be the area bounded by region R..

Q/ Discuss the stability critical point of ODEs 00 X°° + ax + 8 × 3 = 0 B X and draw the phase portrait

Q/Discuss the stability critical point of the ODES X00+6x-x2 + 4X = 0 and draw the phase portrait-

9. Needing a break from studying, you take a walk to the Pogonip koi pond, whereupon a wild-eyed stranger pops out from behind a redwood tree and directs the following polemic in your general direction: "The lies those so-called teachers at that university promulgate, let me tell you. I know the truth that they don't want you to know. As plain as day, " = 0 for all n ≥0. It's an easy induction proof, see?" He hands you a leaflet, where you see the proof that they don't want you to see: We proceed by strong induction on n. Base case: n = 0. We have 10: Induction step: Assume that d1 = = = 0. dx dxk dx = 0 for all kn. Then, by the product rule, nd dx da 1x+1 = 1/1(x²x²) = x²±²x² + x 11 x² d = x.0+x¹.0 0. dx This completes the induction. That derivative rule doesn't seem like the one you learned, but there's nothing obviously wrong with the proof. Is he right, are the math professors propping up the interests of Big Calculus? Or should he have paid better attention in CSE 16? What's going…

Apply Euler's method on the next differential equation with the initial  initial value and in the given interval. You must include: a) table and b) graph.\\\[\frac{d y}{d x}=y^{2}-4 x, \quad y(0)=0.5 ; \quad 0 \leq x \leq 2, \quad \Delta x=0.25\]

7. Define the sequence {b} by bo = 0 Ել ։ = 2 8. bn=4bn-1-4bn-2 for n ≥ 2 (a) Give the first five terms of this sequence. (b) Prove: For all n = N, bn = 2nn. Let a Rsuch that a 1, and let nЄ N. We're going to derive a formula for Σoa without needing to prove it by induction. Tip: it can be helpful to use C1+C2+...+Cn notation instead of summation notation when working this out on scratch paper. (a) Take a a² and manipulate it until it is in the form Σ.a. i=0 (b) Using this, calculate the difference between a Σ0 a² and Σ0 a², simplifying away the summation notation. i=0 (c) Now that you know what (a – 1) Σ0 a² equals, divide both sides by a − 1 to derive the formula for a². (d) (Optional, just for induction practice) Prove this formula using induction.