Univariate Calculus (part 1)

Problem 1

Find  (Underscript[lim , x->∞]) (x^(2 ))sin((1/x))  and  (Underscript[lim , x->(SuperPlus[0])]) (x^(2 ))sin((1/x))

Problem 2

Find (Underscript[lim , n->∞])((Sqrt[(n^2)+4])-n) and (Underscript[lim , n->∞])((Sqrt[(n^2)+n+4])-n)

Problem 3

Find (Underscript[lim , x->∞]) ((sinh(2x-1))/(cosh(2x+3)))

Problem 4

Find (Underscript[lim , n->∞]) (((2^n)n!)/(n^n))

Problem 5

Find (Underscript[lim , n->∞]) (1/n)(Underoverscript[∑ , (k=1), n])sin(((k π)/n))

Problem 6

If (Subscript[a, 1])=3 and (Subscript[a, n+1])=((((Subscript[a, n])^2)-8)/(2(Subscript[a, n]))) for n≥1, what is (Underscript[lim , n->∞])(Subscript[a, n])?

Problem 7

Define f(x)=({{{, ({{(x^2)-2x+3, x∈ℚ}, {0, otherwise}})}}).  For which real numbers a does (Underscript[lim , x->a])f(x) exist?

Problem 8

Find (d/(dx))(((sin^8)((2^x)) )/((((x^2)+9))^12)) and (d/(dx))((tan^(-1))((x^2))+(sec^3)(ln(x)) )

Problem 9

Find (d/(dx))(∫_( cos(x))^( (x^3)))(Sqrt[(t^2)+1])dt

Problem 10

Find the tangent line to (y^2)+2y=(x^3)-5x+2 at the point (1,2).

Problem 11

Find the value of k>0 for which the graphs of (x^2)+(((y-k))^2)=4 and (x^2)-(y^2)=1 are tangent.  Find the point(s) of tangency.

Problem 12

If f(x)=(x^3)+3x+1, explain why (f^(-1))(x) exists and find the tangent line to y=(f^(-1))(x) at x=15.

Problem 13

If f(x)=(((x-2))^11)(e^(2x-4)), find an expression for the value of (f^((20)))(2).

Problem 14

If f(x)=(e^(x/2))cos(((Sqrt[3])/2)x), find (f^((63)))(x).

Limits and Derivatives

Definition and meaning of limits

Basic limit rules

L’Hopital’s rule and indeterminate forms

Connecting limits to other calculus concepts (Riemann sums, infinite series, etc.)

Derivative rules (including the Fundamental Theorem of Calculus)

Tangent lines/slopes

Higher derivatives