Basic Calculus–Sample Final

1. Compute the derivatives of the following functions:a) f(x) = ex

2.

b) f(x) = (x2 + 1)2(2x− 1)3.c) f(x) = x2

x+1.

d) f(x) = x lnx.

e) f(x) = ln[ (x+1)2(x−1)3

(2x−1)4]

2. Find the equation of the tangent line to y = x2 + 6x− 5 at where x = 2.3. Use implicit differentiation to find dy

dxfor the given values of x and y.

a) x3 + y3 = 9; x = 1, y = 2b) xy − y3 = 4; x = 5, y = 1.4. Find the relative maximum and minimum of f(x) = xe2x.5. Find the average of f(x) = x2 + 1 over [1, 3].6. The rate of growth of a certain cell culture is proportional to its size. In 10 hours a

population of 1 million cells grew to 9 million.a) Find the population function P (t) for this cell culture.b) For what t is P (t) = 3, 000, 000?7. If the product of two real numbers is 2, what is the minimum of the sum of squares of

these two numbers?8. Compute the following antiderivatives:a)

∫x2 − 2 dx.

b)∫e2x + 2

xdx.

c)∫ 10

√x dx.

d)∫ 32

2x3 dx.

9. Let f(x) = x2.a) Use Riemann sum to approximate the area under the graph y = f(x) over the interval

1 ≤ x ≤ 5, with n = 4 and using the right endpoints.b) Find the area under the graph y = f(x) over the interval 1 ≤ x ≤ 5.10. Let f(x) = x3 − 3x.a) Find all relative maximum points and relative minimum points of f(x).b) Find all inflection points of f(x).c) Find where f(x) is increasing.d) Find where f(x) is concave up.e) Sketch y = f(x).11. Find the area between the curves y = x2 and y = 3− 2x.12. Let f(x, y) = x3 + x2e3y + 2y3.a) Find ∂f

∂xand ∂f

∂y.

b) Find ∂2f∂x2 , ∂2f

∂y2and ∂2f

∂x∂y.

13. Let f(x, y) = x3 − y2 − 3x + 4y.a) Find all points (x, y) where f(x, y) has a possible relative maximum or relative mini-

mum.b) Use the second derivative test to determine whether the points you find are relative

maximum points or relative minimum points or neither.14. Use Lagrange multiplier method, find the values of x, y that minimizes x2−y2 subject

to the constraint 2x + 3y − 5 = 0.15. The demand equation for a manufacturer is p = 200 − 3x, and the cost function is

C(x) = 75 + 80x− x2 (0 ≤ x ≤ 40).a) Find the revenue function R(x) and profit function P (x).

b) Find the quantity x that maximizes the profit.