The location of a particle at time t is s(t) = -16t2 + 6t + 20. What is the particle’s velocity at t = 2?

Question

1. What is the tangent line to f(x) = e^2x sin x at (π,0)?
2. What is the tangent line to f(x) = x² + 3x at (-2,-2)?
3. What is Lim (x) / (sin x)?
                     x→0
4. The location of a particle at time t is s(t) = -16t² + 6t + 20. What is the particle’s velocity at t = 2?
5. What is the second derivative of f(x) = (x² + 1)² at (3, 100)?

Answer 1

The location of a particle at time t is s(t) = -16t2 + 6t + 20. What is the particle’s velocity at t = 2?
1. What is the tangent line to f(x) = e2x sin x at (π,0)?
2. What is the tangent line to f(x) = x2 + 3x at (-2,-2)?
3. What is Lim (x) / (sin x)?
x→0
4. The location of a particle at time t is s(t) = -16t2 + 6t + 20. What is the particle’s velocity at t = 2?
5. What is the second derivative of f(x) = (x2 + 1)2 at (3, 100)?

 

 

 

1. The general equation of the tangent line is

y -y1 = m(x-x1)   where m = slope = f'(x)

You need to find f'(x) and then m = f'(π)

You have x1 = π, y1 = 0

Substitute these values into the general equation of the tangent line.

 

2. Follow the same procedure for #1.

 

3. This is a standard trigonometric limit, which you can obtain from any Calculus textbook. The limit is 1.

 

4. To find the particle's velocity at t =2.
The velocity = derivative of the position function .  s'(t)
Find s'(2)

 

5.  f(x) = (x2 + 1)2 at (3, 100)

Find f'(x). You can use the power rule followed by the chain rule to determine this.

Alternatively, since the exponent is small, i.e. 2, expand f(x) = (x² + 1)²  = x⁴ + 2x² + 1

(Expanding f(x) is not recommended if the exponent is large).

You can now easily find f'(x) and then f"(x)

 

I have not given you the answers, except for #3 which is standard. I have merely outline certain procedures that you can follow to solve the problems. Other methods may exist to solve these problems. You need to work out the answers yourself to gain the necessary experience in problem solving.