Spring 2014
Math 151
Exam 1 Review
1
1. Find the following limits:
4𝑥 2 +3𝑥+5
a. lim𝑥→∞
b. lim𝑥→5
−2−𝑥+5𝑥 2
2𝑥 2 −13𝑥+15
c. lim𝑥→∞
𝑥 2 −3𝑥−10
𝑥 2 + 3𝑥
+1−𝑥
1 1
−
𝑥 3
d. lim𝑥→3
, 2𝑥 − 3
𝑥−3
1
e. lim𝑥→0 (𝑥 2 cos 2 + 5)
𝑥
4𝑥 2 +3𝑥+1
f. lim𝑥→−∞
g. lim𝑥→2+
2𝑥
7𝑥 −3
4−𝑥 2
1
1
h. lim𝑥→0+ ( −
𝑥
i. lim𝑥→2 cos(
)
|𝑥|
𝑥−2
𝑥 2 −4
𝜋)
2. Use the limit definition to find the derivative of 𝑓 𝑥 = 2𝑥 + 3
a. Find the equation of the line tangent to 𝑓 𝑥 at the point 𝑥 = 2
3. Given that the graph of the function f passes through the point −1, 4 and the equation
of the line tangent to f at this point is given by 𝑦 = 5 − 3𝑥, find lim𝑥→−1
4. Find the value(s) of x for which the function below is not continuous
𝑓 𝑥 −4
𝑥+1
𝑥 + 2,
𝑥 ≤ −1
𝑥−1
, −1 < 𝑥 < 1
𝑓 𝑥 = 𝑥−1
0,
𝑥=1
−𝑥 2 ,
1<𝑥<3
−2𝑥 − 3,
𝑥≥3
5. Find the parametric equation of a line passing through points 1,3 and −5,1
6. Find all the vertical and horizontal asymptotes of the function. Then state where the
function is discontinuous and in which cases the discontinuity is removable.
a. 𝑓 𝑥 =
𝑥 2 +6𝑥+5
𝑥 2 −3𝑥−4
𝑥 2 +2
b. 𝑓 𝑥 =
3𝑥−6
7. Given that 3𝑥 + 2 ≤ 𝑓 𝑥 ≤ 𝑥 3 + 4, 𝑥 ≥ −2, compute lim𝑥→1 𝑓(𝑥)
8. Find the angle between the vectors 3, 1 and − 2𝑖 + 4𝑗
9. What value of x will make the vectors 𝑥𝑖 + 𝑗 and 4 + 𝑥 𝑖 + 3𝑗 orthogonal?
10. Find the distance of the point 1, 5 from the line 2𝑥 − 3𝑦 = 12
1
3
11. Given 𝑟 𝑡 = 𝑡 2 + 1 𝑖 + 𝑡 2 𝑗
a. Find 𝑟 1 and 𝑟(𝑡 + ℎ)
b. Does the graph pass through the point (3, 8)? At what value of t?
c. Eliminate the parameter to find the Cartesian equation
d. Sketch the graph and map its direction
12. A box is held in place by a cable on a frictionless ramp inclined at an angle of 60 0 to the
horizon. If the mass of the box is 50 kgs, find the magnitude of tension in the cable.