What role does the derivative play in defining slopes, tangent lines, and rates of change?

Q: What are Vertical Tangent Lines?

A: Vertical tangent line:

Q: Find first derivative

A: Derivatives can be defined as rate of change of function with respect to an independent variable. It…

Q: Discuss how the concept of instantaneous velocity v(t) can lead to the mathematical notion of a…

A: Topic:- application of derivatives

Q: The rate of change of atmospheric temperature with respect to altitude is equal to the slope of the…

A: The rate of change of atmospheric temperature with respect to altitude is equal to the slope of the…

Q: Use implicit differentiation to find the slope of the tangent line at the point (1, 1).

A:

Q: Estimate the instantaneous rate of change of y(x) = 2x² + 2 at the point: In other words, choose…

A: To estimate the instantaneous rate of change of yx=2x2+2 at the point: x=-2.

Q: Estimate the slope (slope = rise/run) of the tangent line to the curve.

A: answer is in next step . please give a like !!

Q: The Gateway Arch in Saint Louis is modeled by the curve: y = 210 − 20 cosh ( x/30) . At what height…

A: yx = Height of the Gateway archyx = 210 - 20coshx30         Blue curve---------------------formulae…

Q: Explain how to apply the Second Derivative Test.

A: The second derivative test is used to find the relative maxima or relative minima. The second…

Q: What is the slope of the tangent line to the graph of y=x+1/x at x=2? What is the y-intercept of…

A: According to power rule of differentiation, .

Q: 23. The slope of the tangent line indicates the rate of change of the function, also called the 24.…

A: Since, you have asked multiple question, we will solved the first question for you. If you want any…

Q: What function measures the rate of change of distance?

A: _____ function measures the rate of change of distance

Q: What exactly is a derivative?

A: Derivative actually measures the steepness of the graph of a function at a particular point.Thus…

Q: 3. Sketch the graph of the derivative of the function below.

A: A rough graph can be plotted as values are not given in the given graph.

Q: Explain why the slope of the tangent line can be interpreted as aninstantaneous rate of change

A: The average rate of change of a function fx over the interval x, x+h is the slope of the secant line…

Q: Is the slope of the line tangent to the curve the derivative?

A:

Q: What is curve sketching in application of derivatives?

A: We need to answer what is curve sketching.

Q: Explain how to find the tangent line and normal line equation when the slope is undefined?

A: It is given slope at the point is undefined say the point is (x1, y1).  The equation of tangent at…

Q: Determine the derivative of the given function

A:

Q: The slope of a tangent line at a given point on a curve tells the rate of change for the function at…

A: Let the given curve be f(x) -  Then f'(x) will be the slope of a tangent line at a given…

Q: Find the slope of the tangent line to the curve y2+x2 = y4-2x at point (-2,1)

A:

Q: Find the slope of the tangent line to the graph y = x' - 3x +x at the point (2,–2). Hence, find the…

A:

Q: What is the slope of the tangent line to the graph of y x*-3x +x +2 at x-2? What is the y-coordinate…

A:

Q: Explain the relationships among the slope of a tangent line, theinstantaneous rate of change, and…

A: The slope of a tangent line of f on x is,msec=f(x)-f(a)x-a

Q: Use implicit differentiation to find an equation of the tangent line to the curve

A:

Q: Find the derivative of the cotangent function y=cotx

A:

Q: derivative of

A:

Q: What is a derivative?

A: The derivative of a function can be defined in two ways, one approach is graphical and other…

Q: Acceleration is the first time derivative of velocity

A:

Q: the first derivative.]

A: Given function:

Q: tangent line of the curve

A: The tangent line is the line on the curve which passes through the given point on the curve.

Q: What is the slope of the tangent line to the graph of y-(x-1)/(x+1) at x-1?

A: We have to find

Q: Explain Directional Derivative?

A: Our Aim is to explain Directional Derivatives.

Q: Choose (a) or (b). The derivative does not exist if the tangent line is: (a) horizontal (b)…

A: The derivative at a point is equal to the slope of tangent line to the curve at that point. Now for…

Q: 1. Describe the relationship between the derivative of a function to the slope of the tangent line.

A: Query- Describe the relationship between the derivative of a function to the slope of the tangent…

Q: What is the derivative of * y = arcsec(x')?

A:

Q: What is the slope of the tangent line to the graph of y-(x-1)/(x+1) at x-3?

A:

Q: What is the relationship between the secant line, slope of the tangent line, derivative of a…

A: Solved below

Q: Use LONG METHOD DERIVATIVES to find the slope of the tangent line to the curve: f (x) = V7x at point…

A:

Q: What can be said about the relationship between the graph of a  function and the tangent line at a…

A: The point where the second derivarive of graph is zero is called graph of inflection.

Q: Find the derivative using the definition of the equation.

A:

Q: What is the difference between a derivative and an integration?

A: We have to state the difference between a derivative and integral.

Q: Find the instantaneous rate of change cope of the tangent line to 2 (3,-2) 1

A: Since you have asked multiple question, we will solve the first question for you as per our guide…

Q: What is a tangent line?

A: The tangent line of a function is a straight line that passes (touches) through (not crosses or…

Q: Determine the slope of the tangent line to the curve y=2 – x2 at the point (3,-7)

A: slope of tangent = slope of given curve  step 1  we have to find slope of curve  slope = dy/dx  so…

Q: derivative

A: