At t = 2, dy/dt ≠ 2e2k. Remember, dy/dt = k*exp(kt)nomadreid said:Homework Statement
Given path (x,y) described by dx/dt = 1/(t+1), dy/dt = k*exp(kt), constant k > 0. At t = 2 the tangent is parallel to y = 4x + 3. Find k. The given solution: approx. 0.495 .
Homework Equations
(dy/dt)/(dx/dt) = dy/dx = slope of tangent of path
The Attempt at a Solution
At t=2, (dy/dt)/(dx/dt) = (2e^2k)/(1/3) = 6e^2k = 4 (from y=4x+3), giving k=0.5*ln(2/3), or approx -0.2.
To find the value of k in an AP Calculus BC problem, you will need to use the formula for the nth term of an arithmetic sequence. This formula is: an = a1 + (n-1)d, where an is the nth term, a1 is the first term, and d is the common difference. You will also need to know at least two terms in the sequence, as well as the position of the term you are trying to find (n). Plug in the known values into the formula and solve for k.
Finding k in an AP Calculus BC problem is important because it allows you to determine the general term or the nth term of an arithmetic sequence. This can be used to find any term in the sequence and to identify patterns within the sequence. It is also a fundamental concept in calculus and is used to solve more complex problems.
Yes, k can have a negative value in an AP Calculus BC problem. In an arithmetic sequence, the common difference (d) can be positive or negative, which means that k can also be positive or negative. The value of k will depend on the given terms in the sequence and the position of the term you are trying to find.
Finding k in an AP Calculus BC problem is related to differentiation and integration in calculus because it involves finding the slope or rate of change of a function. In an arithmetic sequence, the common difference (d) represents the slope of the line connecting any two consecutive terms. This is similar to finding the derivative of a function in calculus. Integration, on the other hand, involves finding the area under a curve, which can also be represented by the sum of terms in a sequence.
There are a few shortcuts or tricks that can be used to find k in an AP Calculus BC problem. These include using the formula for the sum of an arithmetic series, using the difference of two squares to simplify equations, and using the properties of arithmetic sequences to identify patterns and make calculations easier. However, it is important to understand the concepts behind these shortcuts and not rely on them blindly, as they may not work for all problems.