Let's Get the Highest Points!

1

100

Find the derivative :

1. a^x      2. e^g(x)

1. ln a (a^x)

2. g' (x) e^g(x)

200

Let L(x) = g (f(2x)), find L' (2)

24

300

f'(x)=(4-x)x^-3 for x >0

Find all intervals on which the graph of f is concave down?

f concave down on 0<x<6.

400

(Calculator)

A sugar ant crawls along the vertical edge of a cereal box with a velocity given by v(t)=2^-t +(t-2)² +(t-2)³-(t-2)⁴ +2, for [0,3]. 

For what intervals is the speed of the sugar ant decreasing?

(0,0.776) and (1.474, 2.103)

500

If gas is pumped into a spherical balloon at the rate of 5 ft³ / min, at what rate is the radius increasing when r= 3 ft.

dv/dt = 5 = 4πr²  dr/dt,  V=4/3 πr³

Find dr/dt

5=4π(3)² dr/dt

dr/dt = 5/36π ft/min

2

100

What is the limit if 2x² / (x+5) as x approaches -5 ?

DNE

200

If P(t) = 2 sin t, then find P¹⁹ (2π/3)

1

300

Find the local linear approximation of f(x) = x³-2x+3 at the point where x=2. Use your approximation to estimate f(2.1), f(1.9), f(1.99).

The local linear approximation: y=10(x-2)+7

f(2.1)=8, f(1.9)=6, f(1.99)=6.9

400

Let f(x) = 1/3 x³ +x² -48x +5. 

For which x-value(s) does f(x) have a relative maximum?

x=-8

500

(Calculator)

Let R be the region bounded by the y-axis and the graphs of y=x³/1+x² and y=4-2x. 

The region R is the base of a solid. For this solid, each cross section perpendicular to the x-axis is a square. Find the volume of the solid.

Volume= ∫^1.4877₀ (4-2x- x³/(1+x²) )² dx = 8.997

3

100

f(x) = 3x+ x²  Find ∫²₀ f(x) dx 

26 / 3

200

For a differentiable function g(x), it is known that g(2) = 7, g'(2) =-3, g''(2) = -5.

Use the tangent line approximation at x=2 to estimate g(2.1).

6.7

300

Suppose that f is an integrable function and that 

∫₀¹ f(X) dx =2, ∫₀² f(x) dx=1, ∫₂⁴ f(x) dx=7

Find ∫₀⁴ f(x) dx

8

400

Determine lim_h→0 (1/(x+h) - 1/x) / h

-1/x²

500

Substitution

∫ tan x dx (Hint: Let u=cos x )

ln |sec x| +C

4

100

d/dx (sec^-1 x)

1/ x(x²-1)^1/2

200

Given f(x) =x² on the interval [-2,1], and find the values of c in the open interval (-2,1). (MVT problem)

c= -1/2

300

Evaluate definite integral by using the Net Change Theorem.

∫⁴₀ |x-3| dx

5

400

Given f(x) = (-18x³ +45)^3/2 find the equation of the tangent line at x=1 in the form y=mx+b, Use the tangent line to approximate f (1.1)

m=-2

Equation : y=-2x+5

L(1.1) = -2(1.1) +5 = 2.8

500

Find f(x) by solving the separable differential equation 

dy/dx = 3x²+1/2y with the initial condition f(1)=4

f(x)= (x³+x+14)^1/2

5

100

Take the Anti-derivatives :

∫ sec² (x) dx

tan (x) +C

200

lim_x-1 ln(x) / x² - 1

​​​​

L' Hospital 's Rule!

ans : 1/2

300

f' (x) = (4-x)x^-3

Find the x-coordinate for the critical point of f. Determine whether the point is a relative max, a relative min.

Critical point: x=4

f has a max at x=4, because f'(x) changes from + to -.

400

Find two positive numbers whose product is 220 and whose sum is as small as possible?

Primary Equation: S=x+y  Domain: x>0, y>0

Secondary Equation: xy=220, y= 220/x

S=x+220/x = 

500

A particle moves along the x-axis with velocity at time t on the interval 

[0,+) given by v(t)= -1+e^1-t

Find all values of t at which the particle changes direction.

v(t)=0 when 1=e^1-t, so t=1.

v(t) > 0 for t<1 and v(t)<0 for t>1.

Therefore, the particle changes direction at t=1.