a. Find the tangent to the curve y=2 tan(πx / 4) at x=1 b. What is the smallest value the slope

step 1 Okay, so we're given the equation of a line, and we're asked to find the tangent to the curve at

A. Find the tangent to the curve y=2 tan(πx / 4) at x=1 b....

Key Concepts

Derivative

The derivative of a function represents its instantaneous rate of change, which is used to determine the slope of the tangent line at any given point on the function’s curve. It is foundational in calculus for understanding how functions behave locally.

Chain Rule

The chain rule is a method for differentiating composite functions, which are functions made up of other functions. It allows us to compute the derivative of a function that is composed of an outer function and an inner function by multiplying the derivative of the outer function (evaluated at the inner function) by the derivative of the inner function.

Tangent Line

The tangent line to a curve at a particular point is the straight line that touches the curve at that point and has the same slope as the curve at that point. The equation of the tangent line is typically found using the point-slope form, based on the function’s point of tangency and the derivative (slope) at that point.

Optimization on an Interval

Optimization involves finding the extreme values (minimum or maximum) of a function within a specified interval. This process generally includes finding the derivative of the function, determining the critical points where the derivative is zero or undefined, and then comparing these values, along with the endpoints of the interval, to determine the smallest or largest value the function (or its derivative) can achieve.

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