Math Problem Statement
If u(t) =
sin(5t), cos(6t), t
and
v(t) =
t, cos(6t), sin(5t)
, use the theorem given below to find d dt
u(t) · v(t)
. Suppose u and v are differentiable vector functions. Then d dt
u(t) · v(t)
= u ′(t) · v(t) + u(t) · v ′(t). don't explain, just give me the answer
Solution
The derivative of is:
Do you want more details or have any questions?
Here are 5 related questions:
- How do you compute the dot product of two vector functions?
- What is the significance of the product rule in vector calculus?
- How do you interpret the result geometrically when dealing with vector functions?
- Can you calculate the second derivative of ?
- How do vector functions apply to physical systems, like motion or forces?
Tip: When differentiating vector functions, apply the product rule to each component individually!
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Math Problem Analysis
Mathematical Concepts
Vector Calculus
Differentiation
Dot Product
Product Rule
Formulas
u(t) · v(t) = u₁(t)v₁(t) + u₂(t)v₂(t) + u₃(t)v₃(t)
d/dt [u(t) · v(t)] = u′(t) · v(t) + u(t) · v′(t)
Theorems
Product Rule for Vector Functions
Suitable Grade Level
College Level (Calculus III or Vector Calculus)
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