# Calculus Questions with Answers (5)

Calculus questions, on tangent lines, are presented along with detailed solutions and explanations.

### Question 1

Find parameter p so that y = 3x is a tangent line to the curve
y = x
2 + p

Solution to Question 1:

• The slope of the tangent is equal to 3. At the point of tangency y ' = 3.
y ' = 2 x = 3
• Solve the equation 2 x = 3 to find the x coordinate of the point of tangency.
x = 3 / 2
• The point of tangency is on the line y = 3 x, therefore the y coordinate of the point of tangency is equal to
y = 3 (3/2) = 9 / 2
• The point of tangency (3 / 2 , 9 / 2) is on the curve
y = x 2 + p and is therefore used to find p as follows

9 / 2 = (3 / 2) 2 + p
• Solve the above for p to obtain
p = 9 / 4

### Question 2

a) Find p so that the curve y = x
3 + 2 x 2 + p x + 3 has one horizontal tangent line only.
b) Find the value of x where this tangent line occurs.

Solution to Question 2:

• a) The horizontal tangent line has a slope equal to zero. We first need to find the first derivative of y and equate to 0.
y ' = 3 x 2 + 4 x + p = 0
• For the above quadratic function to have one solution only, it needs to have its discriminant D equal to zero.
D = 16 - 12 p = 0
• Solve the above for p to find
p = 4 / 3
• b) To find the value of x where the horizontal tangent line occurs, we need to solve the above quadratic equation 3 x 2 + 4 x + p = 0 for x. It had the discriminant equal to zero, hence
x = - 4 / 6 = - 2 / 3

### Question 3

Find p and q so that y = 2 x is a tangent, line at x = 3, to the curve
y = p x
2 + q x + 2

Solution to Question 3:

• The point of tangency is on the tangent line y = 2x, hence its y coordinate can also be calculated and its coordinates given by
(3 , 2(3)) = (3 , 6)
• The above point of tangency is also on the curve
y = p x 2 + q x + 2, hence

6 = p(3) 2 + q (3) + 2
• Let us now find y '
y ' = 2 p x + q
• At x = 3 y ' is equal to the slope of the tangent line y = 2 x. Hence
2 = 2 p (3) + q
• We now solve the two linear equations
9 p + 3 q = 4 and 6 p + q = 2
• to find 9 p + 3 q = 4 and 6 p + q = 2
p = 2 / 9 and q = 2 / 3

### Question 4

Find a and b so that y = a x + b is a tangent line to the curve
y = x
2 + 3 x + 2 at x = 3.

Solution to Question 4:

• Find y ' of the curve
y ' = 2 x + 3
• The slope of the tangent line is equal to a and also to y ' at x = 3, hence
a = 2(3) + 3 = 9
• The point of tangency has x = 3 and is on the curve, hence the y coordinate of the point of tangency is given by
y = (3) 2 + 3(3) + 2 = 20
• The point of tangency is also on the tangent line, hence
20 = 3 a + b
• a has already been calculated, b is equal to
b = - 7

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