这个作业是完成微积分相关的数学题
MAT 137Y: Calculus!
1.下面是某些函数f的导数的图。你知道f在所有地方都是连续的在x = 1时,f(0)= 0,并且f(2)=3。绘制f的图。尽可能精确导数图的定性方面,例如对称性。您不必精确关于导数图的定量方面。
2.定义函数f,如下所示:
f(x)= 2020 + 2020
2019 + 2019。。
。+ 21 + 1X。
查找f0
(1)。证明你的答案。提示:构造相关的功能序列并找到其导数在1使用归纳法。
3.使用sin(t)的连续性,sin(kπ2)表示整数k,identity t∈R,sin(t)2 +成本)2 =1。证明对于单位圆上以(0,0)为中心的任何点(x,y)(即满足等式x2 + y2 = 1),∃t∈R s.t. (cos(t),sin(t))=(x,y)。
4.设a,b∈R和r∈R+。考虑一个以(r,a)为中心的圆。我们知道解决方案等式(x − a)2 +(y − b)2 = r2 describes the circle.
(a) Give two differentiable functions f, g : R → R which satisfy:
i. ∀t ∈ R, (f(t) − a)2 + (g(t) − b)2 = r2.
ii. ∀x, y ∈ R, IF (x − a)2 + (y − b)2 = r2, THEN ∃t ∈ R s.t. f(t) = x, g(t) = y.
iii. ∀t ∈ R, f0(t) 6= 0 or g0(t) 6= 0.
Hint: Use 3.
This is a parameterization of the curve (x − a)2 + (y − b)2 = r2. (f(t), g(t)) can be visualized as tracing out the circle as t varies.
(b) Let (x0, y0) be a point on the circle. Compute dy dx for (x − a)2 + (y − b)2 = r2 at (x0, y0) using implicit differentiation. State where your computation is valid.
(c) Let t0 ∈ R. Compute dy dx for (x − a) 2 + (y − b) 2 = r
2 at (f(t0), g(t0)) using the chain rule between the quantities (y = g, x = f, and t). State where your computation is valid.
(d) Let t0 ∈ R. Confirm that at (f(t0), g(t0)), your results in (b) and (c) are the same.
(e) For what values of k ∈ R is the line y = kx + 1 tangent to the circle with centre (5, 1) and radius 4?