Answers

1. To find the values of the parameters a and c in the parabola with the equation f(x) = ax^2 + c such that the slope of the tangent at x = 1 is 4, you can use the following steps:

   Find the derivative of the function f(x). To do this, you will need to use the power rule for differentiation, which states that the derivative of x^n is nx^(n-1). In this case, the derivative of f(x) is 2ax.

   Substitute x = 1 into the derivative of the function f(x). This will give you the slope of the tangent line at x = 1.

   Set the slope of the tangent line at x = 1 equal to 4. This will give you an equation in terms of a and c.

   Solve the equation for a and c. This will give you the values of the parameters a and c that you are looking for.

   For example, suppose you have the parabola f(x) = ax^2 + c, and you want to find the values of the parameters a and c such that the slope of the tangent at x = 1 is 4. The derivative of f(x) is 2ax, so substituting x = 1 into this equation gives you the slope of the tangent line at x = 1: 2a(1) = 2a. Setting this equal to 4 gives you the equation 2a = 4, which you can solve for a to get a = 2.

   To find the value of c, you can substitute the value of a back into the original equation for the parabola, f(x) = ax^2 + c. For example, if x = 1, then f(1) = (2)(1^2) + c = 2 + c. If you know the value of f(1), you can substitute it into this equation to solve for c.

   For example, suppose you know that f(1) = 6. Substituting this value into the equation f(1) = 2 + c gives you 6 = 2 + c, which you can solve for c to get c = 4.

   Therefore, the values of the parameters a and c in the parabola with the equation f(x) = ax^2 + c such that the slope of the tangent at x = 1 is 4 are a = 2 and c = 4.

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