Calculus Answers

𝑓(𝑥) = 9 𝑙𝑜𝑔 (𝑥) + 12 ln (2𝑥 + 1)

Evaluate the integral of f(x, y, z) = x + z - 3 over the cylinder bounded by x^2 + y^2 = 1, z = 0 and z = 1.

Evaluate the indefinite integral in each item

1. ∫(sin 3x + cos 2x) dx
2. ∫sin³ x dx
3. ∫cos⁵ x dx
4. ∫cos² 1/2(x) dx
5. ∫sin³ x cos³ x dx

Evaluate the indefinite integral in each item

1. ∫e^2x+1dx
2. ∫1/(1+e^x) dx
3. ∫e^2x/(e^x+3) dx
4. ∫4(10^{2x^2}) dx
5. ∫x²/(x³+5) dx

1. Suppose the marginal cost for producing x units of a commodity is given by dC/dx= 32 − 0.04x. If it costs $50 to make 1 unit, find the total cost of making 200 units.
2. Find the function f whose tangent has slope 3x^2 + 6x − 2 and whose graph passes through the point (0,6).
3. The methods of differentiation and antidifferentiation are widely used in estimating various quantities in real-world. In economics, experts use tools like marginal cost and marginal revenue as its basic necessities. Calculus is used in this field in calculating the change in demand for a product, calculating change in its cost price, and estimating the rate of change in its revenue with an increase in its selling price. What do you think will happen if the people hired by the government to do these tasks lack expertise in the said skill?

Find the work done pulling a sled 75𝑚 with a 70𝑁 force if the handle makes an angle of 36° with the ground. 

Find the projection of 𝑖 = (4, −1, 4) on 𝑗 = (−2, 0, 4).

Determine the angle between the following two vectors.

|𝑔| = (1,2,−3)

|h| = (0,−3,4)

Find the horizontal and vertical components of a force of 120𝑁 that acts in a direction forming a 58° with the vertical.

Find the vector between 𝑂(−5, 2, 3) and 𝑃(3, 0, −4) in Cartesian form and calculate the magnitude of the vector.