Derivative of Parametric Functions
The derivative of parametric functions is a crucial concept in calculus, allowing us to analyze the rate of change of quantities that vary with respect to a third parameter. In parametric equations, each variable is expressed as a function of an independent parameter, often denoted as t. Understanding how to find the derivative of such functions enables us to determine instantaneous rates of change, velocities, and slopes of curves traced by parametric equations. This introduction sets the stage for exploring the methodology and applications of differentiating parametrically defined functions, providing a foundational understanding essential in various fields of mathematics and science.
Questions
- How do you differentiate the following parametric equation: # x(t)=1/t, y(t)=1/(1-t^2) #?
- How do you differentiate the following parametric equation: # x(t)=t^2-te^t , y(t)=e^(3t) #?
- Given x = cost y=sin2t, how do you find the dy/dx terms parameter t and find the values parameter t points dy/dx = 0?
- How do you find the equation of the tangent to the curve #x=t^4+1#, #y=t^3+t# at the point where #t=-1# ?
- The parametric equation of a curve are #x=t+e^-t#, #y=1-e^-t#, where t takes all real values. How do you find dy/dx in terms of t, and hence find the value of t, for which the gradient of the curve is 1, giving your answer in logarithmic form?
- What is the derivative of #f(t) = (t +te^t, e^t-t^2+t ) #?
- How do you differentiate the following parametric equation: # x(t)=-t-e^t, y(t)= -2t^2-2t^3e^(t) #?
- What is the derivative of #f(t) = (tsint , t^2-tcost ) #?
- How do you differentiate the following parametric equation: # x(t)=te^-t , y(t)=e^t/t #?
- What is the derivative of #f(t) = (t-lnt, tsint ) #?
- How do you find derivatives of parametric functions?
- How do you differentiate the following parametric equation: # x(t)=t, y(t)=t #?
- How do you find the parametric equations of line that passes through the points (1, 3, 2) and ( -4, 3, 0)?
- How do you differentiate the following parametric equation: # x(t)=t^2-t, y(t)=e^t+cost #?
- How do you differentiate the following parametric equation: # x(t)=t-5, y(t)= t^2#?
- What is the derivative of #f(t) = (tcos^2t , t^2-cost ) #?
- How do you differentiate the following parametric equation: # x(t)=3(t+1)^2+2e^t, y(t)= (t+2)^2+t^2#?
- What is the derivative of #f(t) = (t^3-e^(3t-1) , 3t^2+2e^t ) #?
- What is the derivative of #f(t) = (e^(t^2)-e^t, 2t^2-t ) #?
- What is the derivative of #f(t) = (t/(t+1) , te^(2t-1) ) #?