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Find the instantaneous rate of change of the function f(x) = x2 + 3x + 4 when x = 2 using First Principles. Confirm that your answer is correct using the derivative rules.
Find the instantaneous rate of change of the function f(x) = x2 + 3x + 4 when x = 2 using First Principles. Confirm that your answer is correct using the derivative rules.
An object is projected directly up so that its height in metres at time t seconds can be modelled by
h(t) = -0.5t2 + 9t + 9.1
a. From what height was the object initially projected?
b. What was the initial velocity?
c. Find the velocity and height when t = 9s.
d. When does the object return to its initial height?
Evaluate the following indefinite integrals:
- ∫ (12x^5 − 6𝑥^3 − 4x + 1/2 ) dx
- ∫ (√𝑥 + √3)^2 dx
- ∫ (𝑥^𝑒 − 𝑥/𝑒 + 2𝑒𝑥) dx
- ∫ (2𝑡+1)(2𝑡−1)/2√𝑡 dx
- ∫ 8𝑦^3 −1/2𝑦+1 dx
ii) Find all points on the curve x(x + y2) = y where the tangent line is parallel to the y-axis
i) Find all points on the curve x(x + y2) = y where the tangent line is parallel to the x-axis.
ii) Find all points on the curve x(x + y2) = y where the tangent line is parallel to the y-axis
Denote by 𝑄𝑅 and 𝑆 the projections of the point P=(−2,3,−4) onto the 𝑥𝑦 plane, the 𝑦𝑧 plane, and the 𝑥𝑧 plane, respectively. Which of the following line segments has the greatest length?
The ground floor of a school building rests on the 𝑥𝑦
xy-plane in ℝ
3
R3, and the entrance to the building is at the origin. Each step takes you one unit forward, unless you are climbing stairs, in which case a step takes you one unit forward and one unit upward.
You begin at the entrance of the school, facing in the direction of the positive 𝑥
x-axis. You take seven steps, turn to your left, take eight steps up a stairwell, turn to your left, and take four steps to reach your locker. Where is your locker?
Evaluate the ∫(t³+3t)/(t²+1)dt
Evaluate the integral of (z+2)/(z²+4z) dz from -3 to 2
