Exam 1

Math 1231: Single-Variable Calculus

Question 1: Limits

Question 2: Implicit Function1. Prove there is a root for the equation cosθ=θ in the interval (0, π).

• Define f(θ)=cosθ-θ, f(0)=1, f(π)=-1-π, f(θ) have opposite sign on the boundary points, IVP implies there is a root.

2. Prove there is a root for the equation cosy=y in the interval (0, π).• Define f(y)=cosy-y, f(0)=1, f(π)=-1-π, f(y) have opposite sign on the boundary

points, IVP implies there is a root.3. Prove there is a root for the equation cosx=x in the interval (0, π).

• Define f(x)=cosx-x, f(0)=1, f(π)=-1-π, f(x) have opposite sign on the boundary points, IVP implies there is a root.

Prove there is a root for the equation sqrt(y)=-1+y in the interval (0, 4).• Define f(y)=sqrt(y)+1-y, f(0)=1, f(4)=-1, f(y) have opposite sign on the boundary

points, IVP implies there is a root.

Question 2: Implicit Function

Use parenthesis!!!

Question 2: Implicit Function

Question 2: Implicit Function

Question 2: Implicit Function

Question 2: Implicit Function

Question 3: Continuity and Derivative

f(x) is continuous only if lim x0 f(x)=f(0)=0.• How to show that lim x0 x2sin(1/x) =0?

Question 3: Continuity and Derivative

Squeeze theorem

Question 3: Continuity and Derivative

f(x) is continuous only if lim x0 f(x)=f(0)=0.• How to show that lim x0 x2sin(1/x) =0?

-1 <= sin(1/x) <= 1 -x2 <= x2sin(1/x) <= x2. Note that-x2 and x2 approach 0 as x goes to 0, so does x2sin(1/x).

Question 3: Continuity and Derivative

Question 4: Derivative Rules

Question 5: Related Rates