A sum or difference of two functions Local Max and Local Min. Justify your answer with reference to the graph of this function. Disregard desmos restrictions Disregard desmos restrictions. Set sliders = 1 if needed A product of two functions x intercepts. Justify your answer with calculations. Disregard desmos restrictions. Set sliders = 1 if needed A quotient of two functions Domain and Range, Asymptotes. Justify your answer with reference to your equation. Disregard desmos restrictions Disregard desmos restrictions. Set sliders = 1 if needed A composite function IRC at x=A Choose a value for A in the domain of your function and show full calculations. Disregard desmos restrictions. Set sliders = 1 if needed
The Correct Answer and Explanation is:
Here are the completed sections = A = 1.**
Substitute x = 1 into the derivative c'(x).
c'(1) = 2(1) * cos(1²)
c'(1) = 2 * cos(1)
The value cos(1) refers to the cosine of 1 radian. Using a calculator, cos(1) ≈ 0.5403.
Step 3: Calculate the final value.
c'(1) = 2 * 0.5403
c'(1) ≈ 1.0806
Therefore, the Instantaneous Rate of Change of the function c(x) = sin(x²) at x = 1 is 2cos(1), which is approximately 1.0806. This value represents the slope of the tangent line to the graph of the function at the point (1, sin(1)). of the worksheet, with example functions and a full analysis as requested.
A sum or difference of two functions
Function: h(x) = x² + sin(x)
Local Max and Local Min. Justify your answer with reference to the graph of this function.
- Local Minimum: There is one local minimum at approximately (-0.45, -0.23).
- Local Maximum: There are no local maximums.
Justification: The graph of h(x) = x² + sin(x) resembles the base parabola y = x² but with a continuous, small wave-like oscillation caused by the sin(x) term. To find the exact locations of local extrema, we would analyze the derivative, h'(x) = 2x + cos(x). However, as requested, a graphical analysis is sufficient.
Visually, the graph is always concave up, meaning it curves upwards everywhere, like a bowl. It has no “peaks” or local maximums. It does have a single lowest point, or a “valley,” which is the function’s local (and absolute) minimum. This minimum occurs where the upward slope of 2x perfectly cancels the downward slope of cos(x). By inspecting the graph, this single minimum point is located in the third quadrant, at an approximate x-value of -0.45. Since the function is always curving upwards, it never turns back down to form a local maximum.
A product of two functions
Function: p(x) = (x – 2)(x + 3)
x-intercepts. Justify your answer with calculations.
- x-intercepts: x = 2 and x = -3.
Justification: The x-intercepts are the points where the function’s graph crosses the x-axis. At these points, the function’s value, p(x), is equal to zero. We can find these points by setting the function
