I am doing the AP calculus review, these are the questions I have no Idea on how to do:
1. if 0<= k <=pi/2 and the area under the curve y-cosx from x=k to x=pi/2 is 0.2, then k=
2. let F(x) be an antiderivative of (ln x)^4/x If F(1)=0, then F(9)=
3. What is the equation of the line tangent to the graph of f(x)=x^4+3x^2 at the point where f'(x)=1?
4. the base of a solid is a region in the first quadrant bounded by the x-axis, the y-axis, and the line x+2y=8, as shown abobe ( a graph of right triangle with these point (0,0), (0,4), (8,0) in the first quadrant ). If cross sections of the solid perpendicular to the x-axis are semicircles, what is volume of the solid?
Table
x 2 5 7 8
f(x) 10 30 40 10
5. The function f is continuous on the closed interval [2, 8] and has values that are given in the table above. Using the subintervals [2,5], [5,7], [7,8], what is the trapezoidal approximation of integral of f(x) dx from x=2 to x=8 ?
4.134.041
limx>3
To solve these problems, we'll go through each one step by step:
1. To find the value of k, we need to evaluate the definite integral of the function y - cos(x) from x=k to x=pi/2 and set it equal to 0.2. To integrate y - cos(x), we can use the power rule for integration, which states that the integral of x^n with respect to x is (1/(n+1)) * x^(n+1). Integrating y, we get (1/2)y^2, and integrating cos(x), we get sin(x). The integral becomes (1/2)y^2 - sin(x). Evaluating the definite integral and setting it equal to 0.2, we have (1/2)(1^2) - sin(pi/2) - [(1/2)(k^2) - sin(k)] = 0.2. Simplifying this equation and solving for k will give us the value of k.
2. To find the value of F(9), we can use the Fundamental Theorem of Calculus, which states that if F(x) is the antiderivative of f(x), then the definite integral of f(x) from a to b is equal to F(b) - F(a). In this case, we are given that F(1) = 0, so the definite integral of (ln x)^4/x from 1 to 9 is equal to F(9) - F(1). Using this information, we can evaluate the definite integral and find the value of F(9).
3. To find the equation of the tangent line to the graph of f(x) = x^4 + 3x^2 at the point where f'(x) = 1, we need to find the x-coordinate of that point first. Taking the derivative of f(x) with respect to x, we get f'(x) = 4x^3 + 6x. Setting this equal to 1, we can solve for x. Once we have the x-coordinate, we can find the corresponding y-coordinate by substituting it into the original function f(x). With the coordinates of the point, we can use the point-slope formula to find the equation of the tangent line.
4. To find the volume of the solid, we need to calculate the area of each cross section perpendicular to the x-axis and then integrate it over the appropriate range. Since the cross sections are semicircles, the area of each cross section will be half the area of a full circle with a radius equal to the y-value at each x-coordinate. The x-values for the range are given as 0 to 8, which means the y-values can be obtained by substituting x into the equation of the line x + 2y = 8. Once we have the area for each cross section, we can integrate it over the range and find the volume of the solid.
5. To find the trapezoidal approximation of the integral of f(x) dx from x=2 to x=8, we need to use the formula for the trapezoidal rule. The formula is (b-a)/n * [(f(a) + f(b))/2 + Σ(f(xi))], where a and b are the lower and upper limits of the integral, n is the number of subintervals, f(a) and f(b) are the values of the function at the limits, xi are the x-values within each subinterval, and f(xi) are the corresponding values of the function. In this case, we are given the values of the function f(x) at certain x-values, so we can substitute them into the formula and calculate the trapezoidal approximation.